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    [B. mutlu sumer,_jorgen_fredsoe]_hydrodynamics_aro(book_fi.org) [B. mutlu sumer,_jorgen_fredsoe]_hydrodynamics_aro(book_fi.org) Document Transcript

    • Advanced Series on Ocean Engineering — Volume 26HYDRODYNAMICS AROUNDCYLINDRICAL STRUCTURES Revised Edition B. Mutlu S u m e r Jorgen Fredsoe World Scientific
    • HYDRODYNAMICS AROUNDCYLINDRICAL STRUCTURES Revised Edition
    • ADVANCED SERIES ON OCEAN ENGINEERINGSeries Editor-in-ChiefPhilip L- F Liu (Cornell University)PublishedVol. 9 Offshore Structure Modeling by Subrata K. Chakrabarti (Chicago Bridge & Iron Technical Services Co., USA)Vol. 10 Water Waves Generated by Underwater Explosion by Bernard Le Mehaute and Shen Wang (Univ. Miami)Vol. 11 Ocean Surface Waves; Their Physics and Prediction by Stanislaw R Massel (Australian Inst, of Marine Sci)Vol. 12 Hydrodynamics Around Cylindrical Structures by B Mutlu Sumer and Jorgen Fredsoe (Tech. Univ. of Denmark)Vol. 13 Water Wave Propagation Over Uneven Bottoms Part I — Linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics) Part II — Non-linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics)Vol. 14 Coastal Stabilization by Richard Silvester and John R C Hsu (The Univ. of Western Australia)Vol. 15 Random Seas and Design of Maritime Structures (2nd Edition) by Yoshimi Goda (Yokohama National University)Vol. 16 Introduction to Coastal Engineering and Management by J William Kamphuis (Queens Univ.)Vol. 17 The Mechanics of Scour in the Marine Environment by B Mutlu Sumer and Jorgen Fredsoe (Tech. Univ. of Denmark)Vol. 18 Beach Nourishment: Theory and Practice by Robert G. Dean (Univ. Florida)Vol. 19 Saving Americas Beaches: The Causes of and Solutions to Beach Erosion by Scott L. Douglass (Univ. South Alabama)Vol. 20 The Theory and Practice of Hydrodynamics and Vibration by Subrata K. Chakrabarti (Offshore Structure Analysis, Inc., Illinois, USA)Vol. 21 Waves and Wave Forces on Coastal and Ocean Structures by Robert T. Hudspeth (Oregon State Univ., USA)Vol. 22 The Dynamics of Marine Craft: Maneuvering and Seakeeping by Edward M. Lewandowski (Computer Sciences Corporation, USA)Vol. 23 Theory and Applications of Ocean Surface Waves Part 1: Linear Aspects Part 2: Nonlinear Aspects by Chiang C. Mei (Massachusetts Inst, of Technology, USA), Michael Stiassnie (Technion-lsrael Inst, of Technology, Israel) and Dick K. P. Yue (Massachusetts Inst, of Technology, USA)Vol. 24 Introduction to Nearshore Hydrodynamics by lb A. Svendsen (Univ. of Delaware, USA)
    • Advanced Series on Ocean Engineering — Volume 26 HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES Revised Edition B. Mutlu Sumer Jergen Fredsoe Technical University of Denmark, Denmark fc World ScientificNEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • H O N G K O N G • TAIPEI • CHENNAI
    • Published byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HEBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.Cover: Flow around a marine pipeline placed over a trench during a half wave period, calculated byuse of the discrete vortex method.HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES (Revised Edition)Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,electronic or mechanical, including photocopying, recording or any information storage and retrievalsystem now known or to be invented, without written permission from the Publisher.For photocopying of material in this volume, please pay a copying fee through the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.ISBN 981-270-039-0Printed by Fulsland Offset Printing (S) Pte Ltd, Singapore
    • Preface Flow around a circular cylinder is a classical topic within hydrodynamics.Since the rapid expansion of the offshore industry in the sixties, the knowledge ofthis kind of flow has also attracted considerable attention from many mechanicaland civil engineers working in the offshore field. T h e purpose of t h e present book is • To give a detailed, u p d a t e d description of t h e flow p a t t e r n around cylindrical structures (including pipelines) in the presence of waves a n d / o r current. • To describe t h e impact (lift and drag forces) of t h e flow on the structure. • And finally to describe the possible vibration p a t t e r n s for cylindrical struc- tures. This part will also describe the flow around a vibrating cylinder and the resulting forces. T h e scope does not deviate very much from t h e book by Sarpkaya andIsaacson (1980) entitled "Mechanics of Wave Forces on Offshore Structures".However, while Sarpkaya and Isaacson devoted around 50% of the book to thedrag-dominated regime and around 50% to diffraction, the present book concen-trates mainly on the drag-dominated regime. A small chapter on diffraction isincluded for the sake of completeness. T h e reason for our concentration on thedrag-dominated regime (large i f C - n u m b e r s ) is t h a t it is in this field the mostprogress and development have taken place during the last almost 20 years sinceSarpkaya and Isaacsons book. In the drag-dominated regime, flow separation,vortex shedding, and turbulence have a large impact on the resulting forces. Goodunderstanding of this impact has been gained by detailed experimental investiga-tions, and much has been achieved, also in the way of the numerical modelling,especially during the last 5-10 years, when the computer capacity has exploded. In the book the theoretical and the experimental development is described.In order also to make the book usable as a text book, some classical flow solutionsare included in the book, mainly as examples.
    • vi PrefaceAcknowledgement: T h e writers would like to express their appreciation of the very good scien-tific climate in t h e area offshore research in Denmark. In our country the hydrody-namic offshore research was introduced by professor Lundgren at our institute inthe beginning of the seventies. In the late seventies and in the eighties the researchwas mainly concentrated in the Offshore Department at the Danish Hydraulic In-stitute. Significant contributions to the understanding of pipeline hydrodynamicswere here obtained by Vagner Jacobsen and Mads B r y n d u m , two colleagues whosesupport has been of inestimable importance to us. In 1984 a special grant from the university m a d e it possible to ask oneof the authors (Mutlu Sumer) to join the Danish group on offshore engineeringso that he could convey his experience on fluid forces acting on small sedimentparticles to larger structures. This has been followed up by many grants from the Danish Technical Council ( S T V F ) , first through the F T U - p r o g r a m m e and nextthrough the frame-programme "Marine Technique" (1991-97). T h e present bookis an integrated o u t p u t from all these efforts and grants. T h e book has beentypewritten by Hildur Juncker and the drawings have been prepared by Liselotte Norup, Eva Vermehren, Erling Poder, and Nega Beraki. Our librarian Kirsten Dj0rup has corrected and improved our written English.
    • Credits T h e authors and World Scientific Publishing Co P t e Ltd gratefully acknowl-edge the courtesy of t h e organizations who granted permission to use illustrationsand other information in this book.Fig. 3.4:Reprinted from H. Honji: "Streaked flow a r o u n d an oscillating circular cylinder".J. Fluid Mech., 107:509-520, 1982, with kind permission from Cambridge Uni-versity Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road,Cambridge CB2 2RU, UK.Fig. 3.7:Reprinted from C.H.K. Williamson: "Sinusoidal flow relative to circular cylin-ders". J. Fluid Mech., 155:141-174, 1985, with kind permission from CambridgeUniversity Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road,Cambridge CB2 2RU, UK.Figs. 4.51-4.53:Reprinted from E.-S. Chan, H.-F. Cheong and B.-C. Tan: "Laboratory study ofplunging wave impacts on vertical cylinders". Coastal Engineering, 25:87-107,1995, with kind permission from Elsevier Science, Sara Burgerhartstraat 25, 1055KV Amsterdam, T h e Netherlands.Fig. 5.4b:Reprinted from J.E. Fromm and F.H. Harlow: "Numerical solution of the problemof vortex street development". T h e Physics of Fluids, 6(7):975-982, 1963, with kindpermission from American Institute of Physics, Office of Rights and Permissions,500 Sunnyside Blvd., Woodbury, NY 11797, USA.
    • viii CreditsFig. 5.9:Reprinted from P. Justesen: "A numerical study of oscillating flow around a circu-lar cylinder". J. Fluid Mech., 222:157-196, 1991, with kind permission from Cam-bridge University Press, Publishing Division, T h e Edinburgh Building, Shaftes-bury Road, Cambridge CB2 2RU, UK.Fig. 5.14:Reprinted from T. Sarpkaya, C. Putzig, D. Gordon, X. Wang and C. Dalton:"Vortex trajectories around a circular cylinder in oscillatory plus mean flow". J.Offshore Mech. and Arctic Engineering, 114:291-298, 1992, with kind permissionfrom Production Coordinator, Technical Publishing Department, ASME Interna-tional, 345 East 47th Street, New York, NY 10017-2392, USA.F i g . 5.26:Reprinted from P.K. Stansby and P.A. Smith: "Viscous forces on a circular cylin-der in orbital flow at low Keulegan-Carpenter numbers". J. Fluid Mech., 229:159-171, with kind permission from Cambridge University Press, Publishing Division,T h e Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU, UK.Fig. 8.50:Reprinted from R. King: "A review of vortex shedding research and its applica-tion". Ocean Engineering, 4:141-172, 1977, with kind permission from ElsevierScience Ltd., T h e Boulevard, Langford Lane, Kidlington 0 X 5 1GB, UK.
    • List of symbols T h e main symbols used in the book are listed below. In some cases, thesame symbol was used for more t h a n one quantity. This is t o maintain generallyaccepted conventions in different areas of fluid mechanics. In most cases, however,their use is restricted to a single chapter, as indicated in t h e following list.Main symbolsA amplitude of vibrationsA cross-sectional area of b o d y (Chapter 4)Ar, m a x i m u m value of vibration amplitude amplitude of oscillatory flow, or amplitude of horizontal component of orbital motion acceleration (Chapter 4) distance between discrete vortices in an infinite row of vortices ( C h a p t e r 5)a amplitude of surface elevation (Chapter 7)b amplitude of vertical component of orbital motionC concentration or passive quantity (or temperature)CD drag coefficientCD oscillating component of drag coefficientCL lift coefficientCL oscillating component of lift coefficientCLA lift coefficient corresponding to FyACLT lift coefficient corresponding to FyTCld, Cfn lift force coefficients (drag and inertia components, respectively)CL max lift coefficient corresponding to FL m a x^ Lrms lift coefficient corresponding t o Firms force coefficient corresponding to Fxrms
    • X List of symbolsCM inertia coefficientCm hydrodynamic-mass coefficientCmc hydrodynamic-mass coefficient in currentCs force coefficient corresponding to force /c viscous damping coefficientc wave celerity (Chapter 4, Appendix III)cp pressure coefficientD cylinder diameter (or pipeline diameter)D(f,9) directional spectrumE ellipticity of orbital motionE elasticity modulus (Chapter 11)E mean wave energyEx total energyE& energy dissipated in one cycle of vibrationse gap between cylinder and wall, or clearance between pipeline and seabedF Morison force per unit length of structureF external forceFp drag force per unit length of structureFp oscillating component of drag force per unit length of structureFK Froude-Krylov force per unit height of vertical structureFK,tot total Froude-Krylov force on vertical structureFi lift force per unit length of structureFL oscillating component of lift force per unit length of structureFL max m a x i m u m value of lift force per unit length of structureFhrms root-mean-square value of lift force per unit length of structureFN force component normal to structure, per unit length of structureFT total (resultant) force per unit length of structureFrrms root-mean-square value of total (resultant) force per unit length of structureFj, damping forceFf friction drag per unit length of structureFp form drag per unit length of structureFp,Fm predicted and measured in-line forces, respectively (Chapter 4)FTms root-mean-square value of in-line force per unit length of structureFx,Fy force components in Cartesian coordinate systemFx,tot total force on vertical cylinderFy lift force per unit length of structureFyA m a x i m u m value of lift force away from wall per unit length of structureFyT m a x i m u m value of lift force towards wall per unit length of structureFz lift force per unit length of structureFQ force due to potential flow per unit length of cylinder/ frequency, frequency of vibrations
    • List of symbols xi/ impact force on vertical cylinder due to breaking waves (Chapter 4)fl fundamental lift frequencyf„ u n d a m p e d n a t u r a l frequency (or n a t u r a l frequency)fnc n a t u r a l frequency in currentft frequency of transition waves/„ vortex-shedding frequency/„, frequency of oscillatory flow, frequency of wavesfx frequency of in-line vibrationsfy frequency of cross-flow vibrations in forced vibration experimentsfQ peak frequencyg acceleration due to gravityH wave heightHm m a x i m u m wave heightHrms root-mean-square value of wave heightHs significant wave heightH1/3 significant wave height ( = Hs) h water depth h distance between two infinite rows of vortices (Chapter 5)I inertia moment Iu turbulence intensityi imaginary unit Im imaginary part K diffusion coefficient (or thermal conductivity) Ks stability parameter KC Keulegan-Carpenter number KCr Keulegan-Carpenter number for r a n d o m oscillatory flow ks Nikuradses equivalent sand roughness k cylinder roughness (Chapter 4) k spring constant (Chapters 8-11) k wave number fcr, k{ real and imaginary parts of wave number k L correlation length L wave length (Chapter 6, Appendix III) M mass ratio M overturning moment (Chapter 6) m mass of body, per unit length of structure unless otherwise is stated m hydrodynamic mass, per unit length of structure unless otherwise is statedmc hydrodynamic mass in current, per unit length of structure unless otherwise is statedm„ rjth moment of spectrumN normalized vibration frequency in oscillatory flows or in waves f / fw ( = number of vibrations per flow cycle)N(z) tension ( C h a p t e r 11)
    • xii List of symbolsNL normalized lift frequency, /z,// T O (= number of oscillations in lift per flow cycle)n normal directionP pressure forcePr probability of occurrenceP pressureP probability density function (Chapter 7)P fluctuating pressurePo hydrostatic pressurep+ excess pressure1 spectral width parameter9o speedR autocovariance function (Chapter 7)R correlationRe Reynolds numberRer Reynolds number for random oscillatory flowr,e polar coordinatesr,6 spherical coordinates (in axisymmetric flow) (Chapter 5)ro cylinder radiusro sphere radius (Chapter 5)St Strouhal numberS(f) spectrum function of surface elevation (wave spectrum)Sa(f) spectrum function of accelerationSFAI) force spectrumSu(f) spectrum function of velocity£,(/) spectrum function of surface elevation (wave spectrum)T period of oscillatory flow, period of wavesTR return periodTc mean crest periodTs significant wave periodTv vortex-shedding periodT period of oscillatory flow, period of wavesTz mean zero-upcrossing periodT mean periodT0 peak periodt timeU outer flow velocity flow velocity component normal to cylinderuN root-mean-square value of resultant velocity rms current velocityuc wall shear stress velocityUf maximum value of oscillatory-flow velocity, maximum value ofum horizontal component of orbital velocityU rms root-mean-square value of horizontal velocity
    • List of symbols xiiiUw wind speedu flow velocity in boundary layeru,v,w velocity components in Cartesian coordinatesu,v infinitesimal disturbances introduced in velocity componentsu velocity vectorV volume of bodyVm m a x i m u m value of vertical component of orbital velocityVr reduced velocityVrms root-mean-square value of vertical velocityv speedvr, v$ velocity components in polar coordinates, or spherical coordinates (axisymmetric)WQ,WI complex potentialw complex potentialx streamwise distance, or horizontal distanceXd " d y n a m i c " motionXf forced motionx, y Cartesian coordinatesy distance from wallx, y x- and ^-displacements of structure (Chapter 8-11)z vertical coordinate measured from mean water level upwards (Chapter 6, Appendix III)2 spanwise separation distance, or spanwise distancez complex coordinate, z = x + iy = ree (Chapter 5)3 ratio of Reynolds number to Keulegan-Carpenter numberT circulationI vortex strength, corresponding to zth vortex5 b o u n d a r y layer thickness6 goodness-of-fit parameter (Chapter 4)6 phase difference between incident wave and force (Chapter 6)5 logarithmic decrement (Chapter 8)6* displacement thickness of boundary layerSt time incremente spectral width parameterep 1 for p = 0; 2 for p > 1C total dampingC/ fluid dampingC» structural dampingT] surface elevation6 polar coordinate or spherical coordinate6 wave direction (Chapter 7)K strength of individual vortices in an infinite rowA wave length of wavy trajectory of cylinder towed in still fluidfj, dynamic viscosity
    • XIV List of symbolsv kinematic viscosityp fluid densityffu s t a n d a r d deviation of flow velocityav s t a n d a r d deviation of quantity r/r shear stressT normalized wave period (Chapter 7)TO wall shear stressTW wall shear stress (Chapter 4)4> angular coordinate4> phase difference between cylinder vibration and flow velocity ( C h a p t e r 3)<f> potential function (Chapters 4, 6 and Appendix III)(j>i potential function for incident waves<j>, potential function for scattered (reflected plus diffracted) waves ( C h a p t e r 6)<f>s separation angleif phase delayij} stream function%l> infinitesimal disturbance in stream functionu) angular frequency, also angular frequency of external force (for a vibrating system)u> vorticity defined by to = dv/dx — du/dy ( C h a p t e r 5)Ud d a m p e d n a t u r a l angular frequencyu>dv angular frequency of damped free vibrationsu>„ u n d a m p e d natural angular frequencyu)r,u>t real and imaginary p a r t s of angular frequency 10UJV angular frequency of u n d a m p e d free vibrationsoverbar time averageoverdot differentiation with respect to time
    • ContentsPREFACE vCREDITS viiLIST O F SYMBOLS ix1. F l o w a r o u n d a c y l i n d e r in s t e a d y c u r r e n t 1.1 Regimes of flow around a smooth, circular cylinder 1 1.2 Vortex shedding 6 1.2.1 Vortex-shedding frequency 10 1.2.2 Correlation length 28 References 332. F o r c e s o n a c y l i n d e r in s t e a d y c u r r e n t 2.1 Drag and lift 37 2.2 Mean drag 40 2.3 Oscillating drag and lift 50 2.4 Effect of cross-sectional shape on force coefficients 52 2.5 Effect of incoming turbulence on force coefficients 53 2.6 Effect of angle of attack on force coefficients 55 2.7 Forces on a cylinder near a wall 57 References 703. F l o w a r o u n d a c y l i n d e r in o s c i l l a t o r y flows 3.1 Flow regimes as a function of Keulegan-Carpenter number .. 74 3.2 Vortex-shedding regimes 78 3.3 Effect of Reynolds number on flow regimes 89 3.4 Effect of wall proximity on flow regimes 92 3.5 Correlation length 104 3.6 Streaming 116 References 120
    • xvi4. Forces o n a c y l i n d e r in r e g u l a r w a v e s 4.1 In-line force in oscillatory flow 123 4.1.1 Hydrodynamic mass 124 4.1.2 Froude-Krylov force 129 4.1.3 T h e Morison equation 130 4.1.4 In-line force coefficients 133 4.1.5 Goodness-of-fit of the Morison equation 147 4.2 Lift force in oscillatory flow 149 4.3 Effect of roughness 153 4.4 Effect of coexisting current 157 4.5 Effect of angle of attack 161 4.6 Effect of orbital motion 163 4.6.1 Vertical cylinder 163 4.6.2 Horizontal cylinder 169 4.7 Forces on a cylinder near a wall 180 4.8 Forces resulting from breaking-wave impact 187 References 2015. M a t h e m a t i c a l a n d n u m e r i c a l t r e a t m e n t o f flow a r o u n d a c y l i n d e r 5.1 Direct solutions of Navier-Stokes equations 210 5.1.1 Governing equations 211 5.1.2 T h e Oseen (1910) and Lamb (1911) solution 211 5.1.3 Numerical solutions 219 5.1.4 Application to oscillatory flow 227 5.2 Discrete vortex methods 233 5.2.1 Numerical simulation of vorticity transport 234 5.2.2 Procedure used in the implementation of discrete vortex method 237 5.2.3 Application areas 242 5.3 Hydrodynamic stability approach 248 References 2666. D i f f r a c t i o n effect. F o r c e s o n large b o d i e s 6.1 Vertical circular cylinder 276 6.1.1 Analytical solution for potential flow around a vertical circular cylinder 276 6.1.2 Total force on unit-height of cylinder 282 6.1.3 Total force over the depth and the overturning moment .... 287 6.2 Horizontal circular cylinder near or on the seabottom. Pipelines 289 References 295
    • xvii7. Forces o n a cylinder in irregular waves 7.1 Statistical t r e a t m e n t of irregular waves 297 7.1.1 Statistical properties of surface elevation 298 7.1.2 Statistical properties of wave height 312 7.1.3 Statistical properties of wave period 315 7.1.4 Long-term wave statistics 318 7.2 Forces on cylinders in irregular waves 319 7.2.1 Force coefficients 319 7.2.2 Force spectra 325 7.2.3 Forces on pipelines in irregular waves 328 7.2.4 Forces on vertical cylinders in directional irregular waves .. 330 References 3308. F l o w - i n d u c e d v i b r a t i o n s o f a free c y l i n d e r in s t e a d y c u r r e n t s 8.1 A summary of solutions to vibration equation 335 8.1.1 Free vibrations without viscous damping 336 8.1.2 Free vibrations with viscous damping 336 8.1.3 Forced vibrations with viscous damping 338 8.2 Damping of structures 342 8.2.1 Structural damping 342 8.2.2 Fluid damping in still fluid 346 8.3 Cross-flow vortex-induced vibrations of a circular cylinder . 353 8.3.1 Fengs experiment 354 8.3.2 Non-dimensional variables influencing cross-flow vibrations 364 8.4 In-line vibrations of a circular cylinder 376 8.5 Flow around and forces on a vibrating cylinder 383 8.5.1 Cylinder oscillating in the cross-flow direction 383 8.5.2 Cylinder oscillating in in-line direction 396 8.6 Galloping 397 8.7 Suppression of vibrations 407 References 4139. F l o w - i n d u c e d v i b r a t i o n s o f a free c y l i n d e r in w a v e s 9.1 Introduction 418 9.2 Cross-flow vibrations 421 9.2.1 General features 423 9.2.2 Effect of mass ratio and stability parameter 432 9.2.3 Effect of Reynolds number and surface roughness 432 9.2.4 Cross-flow vibrations in irregular waves 436 9.3 In-line vibrations 441 9.4 In-line oscillatory motion 443 9.5 Flow around and forces on a vibrating cylinder 445 References 450
    • xviii10. Vibrations of marine pipelines 10.1 Cross-flow vibrations of pipelines 455 10.1.1 Cross-flow vibrations of pipelines in steady current 455 10.1.2 Cross-flow vibrations of pipelines in waves 465 10.2 In-line vibrations and in-line motions of pipelines 471 10.3 Effect of Reynolds number 473 10.4 Effect of scoured trench 479 10.5 Vibrations of pipelines in irregular waves 481 10.6 Effect of angle of attack 486 10.7 Forces on a vibrating pipeline 486 References 49111. M a t h e m a t i c a l modelling of flow-induced vibrations 11.1 T h e steady-current case 497 11.1.1 Simple models 497 11.1.2 Flow-field models 499 11.2 T h e wave case 503 11.3 Integrated models 506 References 510A P P E N D I X I. Force coefficients for various cross-sectional shapes .... 514A P P E N D I X II. Hydrodynamic-mass coefficients for two- and three- dimensional bodies 517A P P E N D I X III. Small amplitude, linear waves 519REFERENCES FOR APPENDICES 521A U T H O R INDEX 522SUBJECT INDEX 527
    • Chapter 1. Flow around a cylinder in steady current1.1 Regimes of flow around a smooth, circular cylinder T h e non-dimensional quantities describing the flow around a smooth circu-lar cylinder depend on the cylinder Reynolds number Re=™ (1.1) vin which D is the diameter of the cylinder, U is the flow velocity, and v is thekinematic viscosity. T h e flow undergoes tremendous changes as the Reynoldsnumber is increased from zero. T h e flow regimes experienced with increasing Re.are summarized in Fig. 1.1. Fig. 1.2, on the other hand, gives the definitionsketch regarding the two different flow regions referred to in Fig. 1.1, namely thewake and the boundary layer. While the wake extends over a distance which iscomparable with the cylinder diameter, D, the boundary layer extends over a verysmall thickness, 6. which is normally small compared with D. T h e boundary layerthickness, in the case of laminar boundary layer, for example, is (Schlichting, 1979)
    • 2 Chapter 1: Flow around a cylinder in steady current No separation. Creeping flow Re<5 b) -c A fixed pair of symmetric vortices 5 < Re < 40 Laminar vortex 40 < Re < 200 street o^s> Transition to turbulence in the wake 200 < Re < 300 e) Wake completely turbulent. 300 < Re < 3x10 A:Laminar boundary layer separation Subcritlcal A:Laminar boundary layer separation 3 x 10 < R e < 3 . 5 x 10" B:Turbulent boundary layer separation;but Critical (Lower transition] boundary layer laminar B: Turbulent boundary 3.5 x 1CT < R e < 1.5 x 10 layer separation;the %£> boundary layer partly laminar partly turbulent Supercritical h) C <£? C: Boundary layer com- 1.5xio < Re < 4 x l 0 pletely turbulent at one side Upper transition 4x10 < R e C: Boundary layer comple- tely t u r b u l e n t a t Transcritical two sides F i g u r e 1.1 Regimes of flow a r o u n d a s m o o t h , circular cylinder in s t e a d y current.
    • Regimes of flow around a smooth, circular cylinder 3 l n=°(7E) /Reand it is seen t h a t 6/D << 1 for Re larger t h a n 0 ( 1 0 0 ) , say. Wake region Boundary layer Incoming flow Figure 1.2 Definition sketch. Now, returning to Fig. 1.1, for very small values of Re no separation occurs.T h e separation first appears when Re becomes 5 (Figs. 1.1a). For the range of the Reynolds number 5 < Re < 40, a fixed pair of vorticesforms in t h e wake of t h e cylinder (Fig. 1.1 b). T h e length of this vortex formationincreases with Re (Batchelor, 1967). W h e n t h e Reynolds number is further increased, the wake becomes unsta-ble, which would eventually give birth to the phenomenon called vortex sheddingin which vortices are shed alternately at either side of the cylinder at a certainfrequency. Consequently, the wake has an appearance of a vortex street (see Fig.1.3d-f). For the range of t h e Reynolds number 40 < Re < 200 the vortex street islaminar (Fig. 1.1c). T h e shedding is essentially two-dimensional, i.e., it does notvary in the spanwise direction (Williamson, 1989). W i t h a further increase in Re, however, transition to turbulence occurs inthe wake region (Fig. l . l d ) . T h e region of transition to turbulence moves towardsthe cylinder, as Re is increased in the range 200 < Re < 300 (Bloor, 1964). Bloor(1964) reports t h a t at Re — 400, the vortices, once formed, are turbulent. Obser-vations show t h a t the two-dimensional feature of the vortex shedding observed inthe range 40 < Re < 200 becomes distinctly three-dimensional in this range (Ger-rard, 1978 and Williamson, 1988); the vortices are shed in cells in the spanwisedirection. (It may be noted t h a t this feature of vortex shedding prevails for allthe other Reynolds number regimes Re > 300. This topic will be studied in somedetail in Section 1.2.2 in the context of correlation length). For Re > 300, the wake is completely turbulent. T h e boundary layer overthe cylinder surface remains laminar, however, for increasing Re over a very wide
    • 4 Chapter 1: Flow around a cylinder in steady current hi i<.- ." f) R e = 1 6 1 Figure 1.3 Appearance of vortex shedding behind a circular cylinder in stream of oil (from Homann, 1936) with increasing Re.
    • Regimes of flow around a smooth, circular cylinder 5range of Re, namely 300 < Re < 3 x 10 s . This regime is known as t h e subcriticalflow regime (Fig. l . l e ) . W i t h a further increase in Re, transition to turbulence occurs in the bound-ary layer itself. T h e transition first takes place at the point where t h e boundarylayer separates, and then the region of transition to turbulence moves upstreamover the cylinder surface towards t h e stagnation point as Re is increased (Figs.l.lf-l.li). In t h e narrow Re b a n d 3 x 10 5 < Re < 3.5 x 10 5 (Fig. l . l f ) the boundarylayer becomes turbulent at the separation point, b u t this occurs only at one sideof the cylinder. So the boundary layer separation is turbulent at one side of thecylinder and laminar at the other side. This flow regime is called the critical (orthe lower transition) flow regime. T h e flow asymmetry causes a non-zero meanlift on the cylinder, as seen from Fig. 1.4. ICJ 1 0.5 0 0 1 2 3 4 5 R e x 10 Figure 1.4 Non-zero mean lift in the critical-flow regime (3 X 10 5 < Re < 3.5 X 10 5 ). Schewe (1983). T h e side at which the separation is turbulent switches from one side to theother occasionally (Schewe, 1983). Therefore, the lift changes direction, as theone-sided transition to turbulence changes side, shifting from one side to the other(Schewe, 1983). T h e next Reynolds number regime is the so-called supercritical flow regimewhere 3.5 x 10 5 < Re < 1.5 x 10 6 (Fig. l . l g ) . In this regime, the boundarylayer separation is turbulent on b o t h sides of the cylinder. However, transitionto turbulence in t h e boundary layer has not been completed yet; the region oftransition to turbulence is located somewhere between t h e stagnation point andthe separation point. T h e b o u n d a r y layer on one side becomes fully turbulent when Re reaches thevalue of about 1.5 X 10 6 . So, in this flow regime, the b o u n d a r y layer is completelyturbulent on one side of the cylinder and partly laminar and partly turbulent on
    • 6 Chapter 1: Flow around a cylinder in steady currentthe other side. This type of flow regime, called the upper-transition flow regime,prevails over the range of Re, 1.5 x 10 6 < Re < 4.5 X 10 6 (Fig. l . l h ) . Finally, when Re is increased so t h a t Re > 4.5 x 10 6 , the b o u n d a r y layerover the cylinder surface is virtually turbulent everywhere. This flow regime iscalled the transcritical flow regime. Regarding the terminology in relation to t h e described flow regimes and alsothe ranges of Re in which they occur, there seems to be no general consensus amongvarious authors (Farell, 1981). T h e preceding classification and the descriptionare mainly based on Roshkos (1961) and Schewes (1983) works. Roshkos workcovered the Reynolds number range from 10 6 to 10 7 , which revealed t h e existenceof the upper transition and the transcritical regimes, while Schewes work, coveringthe range 2.3 x 10 4 < Re < 7.1 x 10 6 , clarified further details of the flow regimesfrom the lower transition to the transcritical flow regimes.1.2 Vortex shedding T h e most important feature of the flow regimes described in t h e previ-ous section is the vortex-shedding phenomenon, which is common to all the flowregimes for Re > 40 (Fig. 1.1). For these values of Re, the b o u n d a r y layer overthe cylinder surface will separate due to the adverse pressure gradient imposed bythe divergent geometry of the flow environment at the rear side of the cylinder.As a result of this, a shear layer is formed, as sketched in Fig. 1.5. As seen from Fig. 1.6, the boundary layer formed along the cylinder containsa significant amount of vorticity. This vorticity is fed into the shear layer formeddownstream of the separation point and causes the shear layer to roll up into avortex with a sign identical to t h a t of the incoming vorticity. (Vortex A in Fig.1.5). Likewise, a vortex, rotating in the opposite direction, is formed at the otherside of the cylinder (Vortex B).M e c h a n i s m of vortex shedding It has been mentioned in the previous section t h a t the pair formed bythese two vortices is actually unstable when exposed to the small disturbancesfor Reynolds numbers Re > 40. Consequently, one vortex will grow larger t h a nthe other if Re > 40. Further development of the events leading to vortex sheddinghas been described by Gerrard (1966) in the following way. T h e larger vortex (Vortex A in Fig. 1.7a) presumably becomes strongenough to draw t h e opposing vortex (Vortex B) across t h e wake, as sketched inFig. 1.7a. T h e vorticity in Vortex A is in t h e clockwise direction (Fig. 1.5b), whilet h a t in Vortex B is in the anti-clockwise direction. T h e approach of vorticity of
    • Vortex shedding 7 a) Stagnation point b) Shear layer Vorticity layer Detailed picture of flow near separationFigure 1.5 The shear layer. The shear layers on both sides roll up to form the lee-wake vortices, Vortices A and B. Boundary layerFigure 1.6 Distribution of velocity and vorticity in the boundary layer. u> is the vorticity, namely u> = kg-
    • 8 Chapter 1: Flow around a cylinder in steady currentthe opposite sign will t h e n cut off further supply of vorticity to Vortex A from itsboundary layer. This is t h e instant where Vortex A is shed. Being a free vortex,Vortex A is then convected downstream by the flow. Following t h e shedding of Vortex A, a new vortex will be formed at thesame side of the cylinder, namely Vortex C (Fig. 1.7b). Vortex B will now playthe same role as Vortex A, namely it will grow in size and strength so t h a t it willdraw Vortex C across the wake (Fig. 1.7b). This will lead to t h e shedding ofVortex B. This process will continue each time a new vortex is shed at one sideof the cylinder where t h e shedding will continue to occur in an alternate mannerbetween the sides of the cylinder. A B b) B Figure 1.7 (a): Prior to shedding of Vortex A, Vortex B is being drawn across the wake, (b): Prior to shedding of Vortex B, Vortex C is being drawn across the wake. T h e sequence of photographs given in Fig. 1.8 illustrates the time develop-ment of the process during the course of shedding process. One implication of the foregoing discussion is t h a t the vortex shedding oc-curs only when the two shear layers interact with each other. If this interactionis inhibited in one way or another, for example by p u t t i n g a splitter plate at thedownstream side of the cylinder between t h e two shear layers, t h e shedding wouldbe prevented, and therefore no vortex shedding would occur in this case. Also, asanother example, if the cylinder is placed close to a wall, t h e wall-side shear layerwill not develop as strongly as the opposing shear layer; this will presumably leadto a weak interaction between the shear layers, or to practically no interaction ifthe cylinder is placed very close to the wall. In such situations, the vortex shed-
    • Vortex shedding 9Figure 1.8 Time development of vortex shedding during approximately two-third of t h e shedding period. Re = 7 X 1 0 3 .
    • 10 Chapter 1: Flow around a cylinder in steady currentding is suppressed. T h e effect of close proximity of a wall on the vortex sheddingwill be examined in some detail later in t h e next section.1.2.1 Vortex-shedding frequency T h e vortex-shedding frequency, when normalized with the flow velocity Uand the cylinder diameter D, can on dimensional grounds be seen to be a functionof the Reynolds number: St = St(Re) (1.3)in which St = (1.4) Uand fv is the vortex-shedding frequency. T h e normalized vortex-shedding fre-quency, namely S i / i s called the Strouhal number. Fig. 1.9 illustrates how theStrouhal number varies with Re, while Fig. 1.10 gives the power spectra corre-sponding to Schewes (1983) d a t a shown in Fig. 1.9. n St 0.4 0.3 0.2 0.1 Re 0.0 I I i I i I I mil I—i I I mil 1 i I I mil — 40 10 10° 10* 10° 10D 10 •I. h Subcrltical t Supc er- Transcritical Laminar Transition | criti cal vortex to t u r b u l e n c e street in t h e wake Critical. Upper or lower Transition transition Figure 1.9 Strouhal number for a smooth circular cylinder. Experimental data from: Solid curve: Williamson (1989). Dashed curve: Roshko (1961). Dots: Schewe (1983).
    • Vortex shedding 11 Subcritical Re = 1.3 x 10^pUD3 Supercritical Re = 7 . 2 x 10 Re = 1.9 x 10" (J) Upper t r a n s i t i o n Re = 3 . 7 x 10 B e g i n n i n g of t r a n s c r i t i c a l Re = 5.9 x 10 6 0.075 Transcritical 0.2 0.4 0.6 Re = 7.1 x 10 fD/UFigure 1.10 Power spectra of the lift oscillations corresponding to Schewes data in Fig. 1.9 (Schewe, 1983).
    • 12 Chapter 1: Flow around a cylinder in steady current T h e vortex shedding first appears at Re = 40. From Fig. 1.9, the sheddingfrequency St is approximately 0.1 at this Re. It then gradually increases as Reis increased and attains a value of about 0.2 at Re = 300, the lower end of thesubcritical flow regime. From this Re number onwards throughout t h e subcriticalrange St remains practically constant (namely, at the value of 0.2). T h e narrow-band spectrum with t h e sharply defined dominant frequencyin Fig. 1.10a indicates t h a t vortex shedding in the subcritical range occurs in awell-defined, regular fashion. As seen from Fig. 1.9, the Strouhal frequency experiences a sudden j u m pat Re = 3 — 3.5 X 10 5 , namely in the critical Re number range, where St increasesfrom 0.2 to a value of about 0.45. This high value of St is maintained over arather large p a r t of the supercritical Re range, subsequently it decreases slightlywith increasing Reynolds number. T h e large increase in St in the supercritical-flow range is explained as fol-lows: in the supercritical flow regime, the b o u n d a r y layer on both sides of the cylin-der is turbulent at t h e separation points. This results in a delay in the boundary-layer separation where the separation points move downstream, as sketched in Fig.1.11. This means t h a t the vortices (now being closer to each other) would interactat a faster rate t h a n in the subcritical flow regime, which would obviously lead tohigher values of the Strouhal number. Laminar separation Turbulent separation in subcritical regime in s u p e r c r i t i c a l r e g i m e Figure 1.11 Sketch showing positions of separation points at different sepa- ration regimes. T h e power spectrum (Fig. 1.10b) at Re = 7.2 x 10 5 , a Reynolds numberwhich is representative for the supercritical range, indicates t h a t in this Re range,too, the shedding occurs in a well-defined, orderly fashion, since t h e power spec-t r u m appears to be a narrow-band spectrum with a sharply defined, dominantpeak. T h e fact t h a t the magnitude of t h e spectrum itself is extremely small (cf.Figs. 1.10a and 1.10b) indicates, however, that the shed vortices are not as strongas they are in t h e subcritical flow regime. An immediate consequence of this,as will be shown later, is t h a t t h e lift force induced by the vortex shedding isrelatively weak in this Re range.
    • Vortex shedding IS T h e Strouhal number experiences yet another discontinuity when Re reachest h e value of about 1.5 x 10 6 . At this Reynolds number, transition to turbulence inone of the b o u n d a r y layers has been completed (Fig. l . l h ) . So, the b o u n d a r y layerat one side of t h e cylinder is completely turbulent and t h a t at the other side ofthe cylinder is partly laminar and partly turbulent, an asymmetric situation withregard to t h e formation of the lee-wake vortices. This situation prevails over thewhole upper transition region (Fig. l . l h ) . Now, t h e asymmetry in the formation ofthe lee-wake vortices inhibits the interaction of these vortices partially, resultingin an irregular, disorderly vortex shedding. This can be seen clearly from thebroad-band spectra in Figs. 1.10c and d. T h e regular vortex shedding is re-established, however, (see the narrow-band power spectra in Fig. l.lOe and f), when Re is increased to values largert h a n approximately 4.5 x 10 6 , namely the transcritical flow regime where theStrouhal number takes t h e value of 0.25 — 0.30 (Fig. 1.9).Effect o f surface r o u g h n e s s For rough cylinders the normalized shedding frequency, namely the Strouhalnumber, should be a function of b o t h Re and the relative roughness St = St{Re, k3/D) (1.5)in which ks is t h e Nikuradses equivalent sand roughness of t h e cylinder surface. st Smooth 0.5 k s / D = 0.75 x 10 0.4 9 x 10 0.3 30 x 10 0.2 0 104 2 5 10S 2 6 10 6 2 5 10 7 Re Figure 1.12 Effect of surface roughness on vortex-shedding frequency. Strouhal number against Reynolds number. Circular cylinder. Achenbach and Heinecke (1981).
    • lit Chapter 1: Flow around a cylinder in steady current Fig. 1.12 illustrates the effect of the relative roughness on the Strouhal num-ber where t h e experimentally obtained St values for various values of ks/D areplotted against Re (Achenbach and Heinecke, 1981). Clearly, t h e effect is signifi-cant. From t h e figure, it is apparent t h a t , for rough cylinders with ks/D > 3 x 1 0 - 3 ,the critical (the lower transition), the supercritical and t h e upper transition flowregimes merge into one narrow region in the St-Re plane, and t h e flow regimeswitches directly to transcritical over this narrow Re range, and this occurs at verylow values of Re number. (The figure indicates for example t h a t , at Re 0.3 x 10 5for kJD = 30 x 1 0 - 3 and at Re £ 1.5 x 10 5 for k3/D = 3 x 1 0 " 3 ) . This result isin fact anticipated, as it is well known t h a t transition to turbulence occurs muchearlier (i.e., at much smaller values of Reynolds number) over rough walls.E x a m p l e 1.1: Nikuradses equivalent sand roughness In practice there exists an extremely wide variety of surface roughnesses,from small protrusions existing in t h e texture of t h e surface itself to extremely largeroughnesses in the form of marine growth such as mussels and acorn barnacles,etc.. Therefore, normally it is not an easy task to relate t h e roughness of thesurface to some typical scale of t h e roughness elements, partly because t h e elementsare quite unevenly distributed. (On a loose sand bed, for example, the roughnessis measured to be 2-3 times the grain diameter). To tackle this problem, theconcept "Nikuradses equivalent sand roughness" has been introduced. T h e ideais to relate any kind of roughness to t h e Nikuradse roughness so t h a t comparisoncan be m a d e on t h e same basis. Very systematic and careful measurements onrough pipes were carried out by Nikuradse (1933), who used circular pipes. Sandwith known grain size was glued on t h e pipe wall inside t h e pipe. By measuringthe flow resistance and velocity profiles, Nikuradse obtained the following velocitydistribution law ^ - = 5 . 7 5 1 o g 1 0 f + 8.5 (1.6)which can be p u t in t h e following form Uf K ksin which u is the streamwise velocity, Uf is the wall shear-stress velocity, K is theK a r m a n constant ( = 0.4), y is the distance from the wall and fc, is t h e height ofthe sand roughness t h a t Nikuradse used in his experiments (a detailed account of
    • Vortex shedding 15the subject is given by Schlichting (1979)). To judge about t h e roughness of aparticular surface, t h e usual practice is first to measure t h e velocity distributionabove the surface in consideration and then, based on this measured velocity dis-tribution u(y), to determine ka, the Nikuradses equivalent sand roughness of thesurface, from Eq. 1.7.Effect o f c r o s s - s e c t i o n a l s h a p e Fig. 1.13 shows t h e Strouhal-number d a t a compiled by Blevins (1977) forvarious non-circular cross sections, while Fig. 1.14 presents t h e Strouhal numbersfor a variety of profile shapes compiled by ASCE Task Committee (1961). Modi,Wiland, Dikshit and Yokomizo (1992) give a detailed account of flow and vortexshedding around elliptic cross-section cylinders. 0.10 I i — i l i i LJJ i i i_d i i L_LJ 10 2 io3 104 10 5 Re Figure 1.13 Effect of cross-sectional shape on vortex-shedding frequency. Strouhal number against Reynolds number. Blevins (1977). As far as t h e large Reynolds numbers are concerned (iZe>10 5 ), the vor-tex formation process is relatively uninfluenced by t h e Reynolds number for t h ecross sections with fixed separation points such as rectangular cylinders. So, theStrouhal number may not undergo large changes with increasing Re for such cross-sectional shapes, in contrast to what occurs in the case of circular cylinders.Effect o f i n c o m i n g t u r b u l e n c e Quite often, the approach flow is turbulent. For example, a cylinder placedon the sea b o t t o m would feel the approach-flow turbulence which is generatedwithin the b o t t o m boundary layer. T h e turbulence in t h e approach flow is alsoan influencing factor with regard to the vortex shedding. T h e effect of turbulence
    • 16 Chapter 1: Flow around a cylinder in steady current Profile Dimensions Value Profile Dimensions Value (mm) of St (mm) of St t=2.0 t=1.0 T50 12.5 J_L 25 1 -50- 12.5| -50 IT t=0.5 t=1.0 -L. J_L 25 12.5L IE 12.5 12.5 I— 2 5 — T 50 IT t=1.0 t=1.0 25 T 50 5( Th-50^ 1 / 50 t=1.5 t=1.0 12^5 | I h—50—H 1 i 25 2 5 (— / t=1.0 t=1.0 ± 25 25 T 50- I |-K25-f-2&-(—25H Figure 1.14 Effect of cross-sectional shape on Strouhal number. Strouhal numbers for profile shapes. ASCE Task Committee (1961).
    • Vortex shedding 17 Profile Dimensions Value Plow Profile Dimensions Value Flow (mm) of St (mm) of St t= L.() 1 t=1.0 D.145 — 12.5 12.5| | 25 Tr—50^| t 0.168 t h-25- 25-| t=1.5 *~ 0.156 t t-l.U 50 0.160 h—50 - H 0.145 1h io n .1 U "| Cylinder t=l. 0 11800 <Re< 19100 *- 0.114 1 0.200 25 25 t t i> Figure 1.14 (continued.)on the vortex shedding has been studied by various authors, for example by Che-ung and Melbourne (1983), Kwok (1986) and Norberg and Sunden (1987) amongothers. Fig. 1.15 presents the Strouhal number data obtained by Cheung andMelbourne for various levels of turbulence in their experimental tunnel. Here, Iuis the turbulence intensity defined by h= (1-in which V u 2 is the root-mean-square value of the velocity fluctuations and u isthe mean value of the velocity. The variation of St with the Reynolds number changes considerably withthe level of turbulence in the approach flow. The effect of turbulence is rathersimilar to that of cylinder roughness. The critical, the supercritical, and the uppertransition flow regimes seem to merge into one transitional region.
    • 18 Chapter 1: Flow around a cylinder in steady current n st 0.4- 0.3" Mt. Isa stack full scale data for Iu = 7.8% o.i- St = 0.20 at Re = 4 x 10 10 St = 0.15 at Re = 2 x 10 ? Re 10" Figure 1.15 Effect of turbulence in the approach flow on vortex-shedding frequency. Strouhal numbers as a function of Reynolds number for different turbulence intensities. Iu is the level of turbulence (Eq. 1.8). Cheung and Melbourne (1983). It appears from the figure t h a t the lower end of this transition range shiftstowards the smaller a n d smaller Reynolds numbers with t h e increased level ofturbulence. This is obviously due to the earlier transition to turbulence in thecylinder boundary layer with increasing incoming turbulence intensity.Effect o f s h e a r in t h e i n c o m i n g flow T h e shear in the approach flow is also an influencing factor in the vortexshedding process. T h e shear could be present in the approach flow in two ways:it could be present in the spanwise direction along the length of the cylinder (Fig.1.16a), or in the cross-flow direction (Fig. 1.16b). T h e characteristics of shear flowaround bluff bodies including t h e non-circular cross-sections have been reviewedby Griffin (1985a and b). In t h e case when the shear is present in the spanwisedirection (Fig. 1.16a), the vortex shedding takes place in spanwise cells, with a
    • Vortex shedding 19frequency constant over each cell. Fig. 1.17 clearly shows this; it is seen t h a t theshedding occurs in four cells, each with a different frequency. W h e n t h e Strouhalnumber is based on the local velocity (the dashed lines in the figure), t h e d a t a aregrouped around t h e Strouhal number of about 0.25. a) b) Figure 1.16 Two kinds of shear in the approach flow, a: Shear is in the spanwise direction, b: Shear is in the cross-flow direction. Regarding the length of cellular structures, research shows that the lengthof cells is correlated with t h e degree of t h e shear. T h e general t r e n d is t h a t t h ecell length decreases with increasing shear (Griffin, 1985a). W h e n the shear takes place in the cross-stream direction (the conditions inthe spanwise direction being uniform), the shedding is only slightly influenced forsmall and moderate values of the shear steepness s which is defined by D_du (1.9) Uc dyFor large values of s, however, the shedding is influenced somewhat substantially(Kiya, T a m u r a and Arie, 1980). Fig. 1.18 shows the Strouhal number plottedagainst the Reynolds number for three different values of s. As is seen for s = 0.2,t h e Strouhal number is increased substantially relative t o t h e uniform-flow case(s = 0).
    • 20 Chapter 1: Flow around a cylinder in steady current Stii ID U,, 0.32 0.28 - fD 0.24- 0.20 i I I I i -z/D 0 2 10 14 18 Figure 1.17 Effect of shear in the approach flow on vortex-shedding fre- quency. Shear in the spanwise direction. Circles: Strouhal num- ber based on the centre-line velocity Uc. Dashed lines: Strouhal number based on the local velocity, UociLi. Re = 2.8 X 10 4 . The shear steepness: s = 0.025. Maull and Young (1973).S -g Re 10 Figure 1.18 Effect of shear in the approach flow frequency. Shear in cross- flow direction. The Strouhal number against the Reynolds num- ber for three different values of the shear steepness s. Hatched band: Uniform-flow results. Circles: Shear-flow results. Kiya et al. (1980).
    • Vortex shedding 21Effect of w a l l p r o x i m i t y This topic is of direct relevance with regard to pipelines. W h e n a pipeline isplaced on an erodible sea bed, scour may occur below the pipe due to flow action.This may lead to suspended spans of the pipeline where the pipe is suspendedabove the bed with a small gap, usually in t h e range from 0 ( 0 . I D ) to 0 ( 1 D ) .Therefore it is important to know what kind of changes take place in the flowaround and in the forces on such a pipe. 777777777777777777777 Figure 1.19 Flow around a) a free cylinder, b) a near-wall cylinder. S = separation points. W h e n a cylinder is placed near a wall, a number of changes occur in theflow a r o u n d t h e cylinder. These changes are summarized as follows: 1) Vortex shedding is suppressed for the gap-ratio values smaller t h a n aboute/D = 0.3, as will be seen later in the section. Here, e is the gap between thecylinder and the wall. 2) T h e stagnation point moves to a lower angular position as sketched inFig. 1.19. This can be seen clearly from the pressure measurements of Fig. 2.20aand Fig. 2.20b where t h e mean pressure distributions around t h e cylinder aregiven for three different values of t h e gap ratio. While the stagnation point islocated at about <f> = 0° when e/D = 1, it moves to the angular position of about<j> = —40° when the gap ratio is reduced to e/D = 0.1. 3) Also, the angular position of the separation points changes. T h e sepa-ration point at t h e free-stream side of t h e cylinder moves u p s t r e a m and t h a t atthe wall side moves downstream, as shown in t h e sketch given in Fig. 1.19. T h e
    • 22 Chapter 1: Flow around a cylinder in steady current Free-stream side 140 separation point (a) 120 100 77777777777777- 8 0 *»>*- 60 (b) 140 120 7777^77777777 Wall s i d e 100 separation point 80 60 i i i i i J_l I I I I L 0 1 e/D Figure 1.20 Angle of separation as a function of the gap ratio, (a): At the free-stream side of the cylinder and (b): At the wall side of the cylinder. ije = 6 x 10 3 . Jensen and Sumer (1986).separation angle measured for a cylinder with Re = 6 x 10 3 is shown in Fig. 1.20;the figure indicates t h a t for example for e/D = 0.1 the separation angle at thefree-stream side is (j> = 80°, while it is rf> = —110° at the wall side for the same gapratio. 4) Finally, the suction is larger on the free-stream side of the cylinder t h a non the wall-side of the cylinder, as is clearly seen in Fig. 2.20b and c. W h e n thecylinder is placed away from the wall, however (Fig. 2.20a) this effect disappearsand the symmetry is restored.
    • Vortex shedding 23 At A At B -2.0 -2.0 =2 -®: -3.0 -4.0 O 0.8 1.6 2.4 0.8 1.6 2.4 ///*//// Log. frequency (Hz) Log. frequency (Hz) c -2.0 -2.0b) 0.3 -3.0 -3.0 a (0 oio o -4.0 V -4.0 0.8 1.6 2.4 0 0.8 1.6 2.4 Log. frequency (Hz) Log. frequency (Hz) C I a; XI - -2.4c) 0.2 y v -3.2 a. to oio -4.0 o 0.8 1.6 2.4 0.8 1.6 2.4 Log. frequency (Hz) Log. frequency (Hz)d) i -2.4 -a 1 -3.2 s. -4.0 _i i i i_ CD 0 0.8 1.6 2.4 •ats Log. frequency (Hz) 2 Figure 1.21 Effect of wall proximity on vortex shedding. Power spectra of the hot-wire signal received from the wake. Bearman and Zdravkovich (1978).
    • 24 Chapter 1: Flow around a cylinder in steady current Vortex shedding may be suppressed for a cylinder which is placed close to awall. Fig. 1.21 presents power spectra of the hot-wire signals received from b o t hsides of the wake of a cylinder placed at different distances from a wall (Bearmanand Zdravkovich, 1978). As is clearly seen, regular vortex shedding, identified bythe sharply defined, dominant peaks in the power spectra, persists only for valuesof the gap-to-diameter ratio e/D down to about 0.3. This result, recognized firstby B e a r m a n and Zdravkovich, was later confirmed by the measurements of Grass,Raven, Stuart and Bray (1984). T h e photographs shown in Fig. 1.22 demonstratethe supression of vortex shedding for gap ratios e/D below 0.3. T h e suppression of vortex shedding is linked with the asymmetry in thedevelopment of the vortices on the two sides of the cylinder. T h e free-stream-sidevortex grows larger and stronger t h a n the wall-side vortex. Therefore the interac-tion of the two vortices is largely inhibited (or, for small e/D, totally inhibited),resulting in partial or complete suppression of the regular vortex shedding. Regarding the effect of wall proximity on the vortex-shedding frequency forthe range of e/D where the vortex shedding exists, measurements show t h a t theshedding frequency tends to increase (yet slightly) with decreasing gap ratio. InFig. 1.23 are plotted t h e results of two studies, namely Grass et al. (1984) andRaven, Stuart, Bray and Littlejohns (1985). Grass et al.s experiments were donein a laboratory channel with b o t h smooth and rough beds. T h e surface of the testcylinder was smooth. Their results collapse onto a common curve when plottedin the normalized form presented in the figure where Stg is the Strouhal numberfor a wall-free cylinder. T h e d a t a points of Raven et al.s study, on the otherhand, were obtained in an experimental program conducted in the Severn Estuary(UK) where a full-scale pipeline (50.8 cm in diameter with a surface roughnessof k/D = 8.5 x 1 0 - 3 ) was used. In both studies, St is defined by the veloc-ity at the top of the cylinder. There are other d a t a available such as Bearmanand Zdravkovich (1978) and Angrilli, Bergamaschi and Cossalter (1982). WhileBearman and Zdravkovichs measurements indicate t h a t the shedding frequencypractically does not change over t h e range 0.3 < e/D < 3, Angrilli et al.s mea-surements show t h a t there is a systematic (yet, slight) increase in the sheddingfrequency with decreasing gap ratio in their measurement range 0.5 < e/D < 6(they report a 10% increase in the shedding frequency at e/D = 0.5). It is apparent from the existing d a t a t h a t the vortex-shedding frequency isinsensitive to the gap ratio, although there seems to be a tendency t h a t it increasesslightly with decreasing gap ratio. This slight increase in the Strouhal frequencymay be a t t r i b u t e d to the fact t h a t the presence of the wall causes the wall-sidevortex to be formed closer to the free-stream-side vortex. As a result of this, thetwo vortices interact at a faster rate, leading to a higher St frequency.
    • Vortex shedding 25 a)^o0.4 b) = 0.3 c) = 0.2 d) = 0.05Figure 1.22 Effect of wall proximity on vortex shedding. Flow in the wake of a near-wall cylinder. Shedding is apparent for e/D = 0.4 and 0.3 but suppressed for e/D = 0.2and 0.05. Re = 7 x l 0 3 .
    • 26 Chapter 1: Flow around a cylinder in steady current e/D Figure 1.23 Effect of wall proximity on vortex shedding frequency. Nor- malized Strouhal number as a function of gap ratio. St0 is the Strouhal number for wall-free cylinder. Circles: Raven et al. (1985). Solid curve: Grass et al. (1984). Jensen, Sumer, Jensen and Freds0e (1990) investigated t h e flow around apipeline (placed initially on a flat bed) at five characteristic stages of the scourprocess which take place underneath the pipeline. Each stage was characterized inthe experiments by a special, frozen scoured bed profile, which was an exact copyof the measured bed profile of an actual scour test. T h e investigated scour profilesand the corresponding mean flow field are shown in Fig. 1.24. It was observedthat no vortex shedding occurred for the first two stages, namely stages I and II,while vortex shedding did occur for stages III - V. Fig. 1.25 depicts t h e sheddingfrequency corresponding to the different stages. T h e variation of the Strouhal number, which goes from as high a value as0.36 for Stage III to an equilibrium value of 0.17 in Stage V, can be explained bythe geometry of the downstream scour profile as follows. For profiles III and IV, the steep slope of t h e u p s t r e a m p a r t of the dunebehind the cylinder forces the shear layer originating from t h e lower edge of thecylinder to bend upwards, thus causing t h e associated lower vortex to interactwith the upper one prematurely, leading to a p r e m a t u r e vortex shedding. T h eresult of this is a higher vortex shedding frequency and a very narrow formationregion. T h e flow visualization study carried out in the same experiments (Jensenet al., 1990) confirmed the existence of this narrow region.
    • Vortex shedding 27 y(cm) 6* 4-1 20 c m / s - 2 I ///>//////////•/?//////?////////////////////;;;/;//;/ S )- j H j J II 1 ft?V s - - * • : in yx^-/"/"X//^[ J b > / /////////////////////////// 1 i —"i 1 r- - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8 x/DFigure 1.24 Vector plot of the mean velocities, S = the approximate posi- tion of the stagnation point. Jensen et al. (1990). A St 0.4- 0.2 , time (mln) 1 10 100Figure 1.25 Time development of Strouhal number during the scour process below a pipeline. Jensen et al. (1990).
    • 28 Chapter 1: Flow around a cylinder in steady current1.2.2 Correlation length As has been mentioned in Section 1.1, vortex shedding in the turbulentwake regime (i.e. iJe>200) occurs in cells along the length of the cylinder. These spanwise cell structures are visualized in Fig. 1.26 which shows thetime evolution of the shedding process in plan view. T h e cells are quite clear from the photographs in Fig. 1.26. Shedding doesnot occur uniformly along the length of the cylinder, b u t rather in cells (designatedby A, B and C in Fig. 1.26). It can also be recognized from the pictures in Fig.1.26 that the cells along the length of the cylinder are out of phase. Consequently,the maximum resultant force acting on the cylinder over its total length may besmaller t h a n the force acting on the cylinder over the length of a single cell. T h e average length of the cells may be termed t h e correlation length. T h eprecise determination of the correlation length requires experimental determina-tion of the spanwise variation of t h e correlation coefficient of some unsteady quan-tity related to vortex shedding, such as fluctuating surface pressure, or a fluctuat-ing velocity just outside the shear layer at separation. T h e correlation coefficient is defined by R(z) = X^-P^ + Z) (1.10)in which £ is the spanwise distance, z is the spanwise separation between twomeasurement points, and p is the fluctuating part of the unsteady quantity inconsideration. T h e overbar denotes the time averaging. T h e correlation length L,on the other hand, is defined by the integral /•oo L= / R(z)dz (1.11) Jo Fig. 1.27 gives a typical example of the correlation coefficient obtained in awind tunnel with a cylinder 7.6 cm in diameter and 91.4 cm in length with largestreamlined end plates (Novak and Tanaka, 1977). T h e Reynolds number was1.9 x 10 4 . T h e measured quantity was the surface pressure at an angle 60° to themain stream direction. T h e correlation length corresponding to the correlationcoefficient, given in Fig. 1.27, on the other hand is found to be L/D = 3 from Eq.1.11.
    • t = 0 0.3 s 0.5 s 0.9 s U Uu— a) b) c) . d)Figure 1.26 Photographs, illustrating the time evolution of spanwise cell structure. Cyli
    • SO Chapter 1: Flow around a cylinder in steady current For a smooth cylinder, the correlation length changes with the Reynoldsnumber. Table 1.1 presents the correlation-length d a t a compiled by King (1977). Table 1.1 Correlation lengths and Reynolds numbers of smooth cylin- ders. Reynolds number Correlation Source length 40 < Re < 150 (15-20)D Gerlach and Dodge (1970) 150 < Re < 105 (2-3)D Gerlach and Dodge (1970) 1.1 x 104 < Re < 4.5 x 104 (3-6)D El-Baroudi (1960) > 105 0.5D Gerlach and Dodge (1970) 2 x 105 1.56D Humphreys (1960) T h e table shows t h a t the correlation length is (15-20)D for 40 < Re < 150but experiences a sudden drop to (2-3)D at Re = 150. T h e latter Re number isquite close to the Reynolds number (see Fig. l . l d ) , at which t h e laminar vortexshedding regime disappears. Regarding the finite (although large) values of thecorrelation length in the range 40 < Re < 150, the correlation length in this flowregime should theoretically be infinite, since the vortex regime in this range isactually two-dimensional. However, purely two-dimensional shedding cannot beachieved in practice due to the existing end conditions. A slight divergence fromthe purely two-dimensional shedding, in the form of the so-called oblique shedding(see for example Williamson, 1989), may result in finite correlation lengths. Other factors also affect the correlation. T h e correlation increases consider-ably when the cylinder is oscillated in the cross-flow direction. Fig. 1.28 presentsthe correlation coefficient d a t a obtained by Novak and Tanaka (1977) for severalvalues of t h e double-amplitude-diameter ratio 2A/D where A is the amplitude ofcross-flow vibrations of t h e cylinder. T h e figure shows t h a t the correlation coef-ficient increases tremendously with t h e amplitude of oscillations. Similar resultswere obtained by Toebes (1969) who measured the correlation coefficient of fluctu-ating velocity in the wake region near the cylinder. Fig. 1.29 presents the variationof the correlation length as a function of the amplitude-to-diameter ratio (curve ain Fig. 1.29). Clearly, the correlation length increases extensively with increasingthe amplitude of oscillations.
    • Vortex shedding SI z/D Figure 1.27 Correlation coefficient of surface pressure fluctuations as func- tion of the spanwise separation distance z. Cylinder smooth. Re = 1.9 X 10 4 . Pressure transducers are located at 60° to the main stream direction. Novak and Tanaka (1977). R i i 1.0- 2A/D = 0.20 = 0.8- 0.6- 0.15 0.4- «V*. ^^>~~ o— 0 0.10 0.2- 0.05 A • " - • ^0~^~ 0 0H 1 i 1 1 * 1 * z/D (3 2 4 6 8 10 Figure 1.28 Effect of cross-flow vibration of cylinder on correlation co- efficient of surface pressure fluctuations. Cylinder smooth. Re = 1.9 X 10 . Pressure transducers are located at 60° to the main stream direction. A is the amplitude of the cross-flow vibrations of cylinder. Novak and Tanaka (1977). Turbulence in the approaching flow is also a significant factor for the corre-lation length, as is seen from Fig. 1.29. T h e turbulence in the tests presented inthis figure was generated by a coarse grid in the experimental tunnel used in Novakand Tanakas (1977) study. T h e figure indicates t h a t t h e presence of turbulence
    • S2 Chapter 1: Flow around a cylinder in steady currentin the approaching flow generally reduces the correlation length. It is interestingto note t h a t with 2A/D = 0.2, while the correlation length increases from about 3diameters to 43 diameters for a turbulence-free, smooth flow, the increase is not sodramatic when some turbulence is introduced into the flow; t h e correlation lengthincreases to only about 10 diameters in this latter situation. D 40- Flow: 30- a: S m o o t h 20- b: Turbulent 10- 0 -£ 1 1 *- 2 A / D 0 0.1 0.2 Figure 1.29 Correlation length. Cylinder smooth. Re = 1.9 X 10 4 . Pres- sure transducers are located at 60° to the main stream direc- tion. A is the amplitude of cross-flow vibrations of the cylinder. Turbulence in the tunnel was generated by a coarse grid, and its intensity, Iu = 11%. Novak and Tanaka (1977). T h e subject has been most recently studied by Szepessy and Bearman(1992). These authors studied the effect of the aspect ratio (namely the cylin-der length-to-diameter ratio) on vortex shedding by using moveable end plates.They found t h a t the vortex-induced lift showed a m a x i m u m for an aspect ratio of1, where the lift could be almost twice the value for very large aspect ratios. Thisincrease of the lift amplitude was found to be accompanied by enhanced spanwisecorrelation of the flow. Finally, it may be noted t h a t Ribeiro (1992) gives a comprehensive reviewof the literature on oscillating lift on circular cylinders in cross-flow.
    • References SSREFERENCESAchenbach, E. and Heinecke E. (1981): On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 x 10 3 to 5 x 10 6 . J. Fluid Mech., 109:239-251.Angrilli, F., Bergamaschi, S. and Cossalter, V. (1982): Investigation of wall- induced modifications to vortex shedding from a circular cylinder. Trans. of the ASME, J. Fluids Engrg., 104:518-522.ASCE Task Committee on W i n d Forces (1961): W i n d forces on structures. Trans. ASCE, 126:1124-1198.Batchelor, G.K. (1967): An Introduction to Fluid Dynamics. Cambridge Univer- sity Press.Bearman, P.W. and Zdravkovich, M.M. (1978): Flow around a circular cylinder near a plane boundary. J. Fluid Mech., 89(l):33-48.Blevins, R.D. (1977): Flow-induced Vibrations. Van Nostrand.Bloor, M.S. (1964): T h e transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 19:290-304.Cheung, J.C.K. and Melbourne, W.H. (1983): Turbulence effects on some aerody- namic parameters of a circular cylinder at supercritical Reynolds numbers. J. of W i n d Engineering and Industrial Aerodynamics, 14:399-410.El-Baroudi, M.Y. (1960): Measurement of Two-Point Correlations of Velocity near a Circular Cylinder Shedding a K a r m a n Vortex Street. University of Toronto, UTIAS, T N 3 1 .Farell, C. (1981): Flow around fixed circular cylinders: Fluctuating loads. Proc. of ASCE, Engineering Mech. Division, 107:EM3:565-588. Also see the closure of t h e paper. Journal of Engineering Mechanics, ASCE, 109:1153-1156, 1983.Gerlach, C.R. and Dodge, F . T . (1970): An engineering approach to t u b e flow- induced vibrations. Proc. Conf. on Flow-Induced Vibrations in Reactor System Components, Argonne National Laboratory, pp. 205-225.Gerrard, J.H. (1966): T h e mechanics of the formation region of vortices behind bluff bodies. J. Fluid Mech., 25:401-413.
    • 34 Chapter 1: Flow around a cylinder in steady currentGerrard, J.H. (1978): T h e wakes of cylindrical bluff bodies at low Reynolds number. Phil. Transactions of the Royal Soc. London, Series A, 288(A1354):351-382.Grass, A.J., Raven, P.W.J., Stuart, R.J. and Bray, J.A. (1984): T h e influence of b o u n d a r y layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. Trans. ASME, J. of Energy Res. Technology, 106:70-78.Griffin, O.M. (1985a): Vortex shedding from bluff bodies in a shear flow: A Re- view. Trans. ASME, J. Fluids Eng., 107:298-306.Griffin, O.M. (1985b): T h e effect of current shear on vortex shedding. Proc. Int. Symp. on Separated Flow Around Marine Structures. T h e Norwegian Inst. of Technology, Trondheim, Norway, J u n e 26-28, 1985, p p . 91-110.Homann, F . (1936): Einfluss grosser Zahigkeit bei Stromung u m Zylinder. Forschung auf dem Gebiete des Ingenieurwesen, 7(1):1-10.Humphreys, J.S. (1960): On a circular cylinder in a steady wind at transition Reynolds numbers. J. Fluid Mech., 9:603-612.Jensen, B.L. and Sumer, B.M. (1986): Boundary layer over a cylinder placed near a wall. Progress Report No. 64, Inst, of Hydrodynamics and Hydraulic Engineering, ISVA, Techn. Univ. Denmark, p p . 31-39.Jensen, B.L., Sumer, B.M., Jensen, H.R. and Freds0e, J. (1990): Flow around and forces on a pipeline near a scoured bed in steady current. Trans, of t h e ASME, J. of Offshore Mech. a n d Arctic Engrg., 112:206-213.King, R. (1977): A review of vortex shedding research and its application. Ocean Engineering, 4:141-171.Kiya, M., Tamura, H. and Arie, M. (1980): Vortex shedding from a circular cylin- der in moderate-Reynolds-number shear flow. J. Fluid Mech., 141:721-735.Kwok, K.C.S. (1986): Turbulence effect on flow around circular cylinder. J. En- gineering Mechanics, ASCE, 112(11):1181-1197.Maull, D.J. and Young, R.A. (1973): Vortex shedding from bluff bodies in a shear flow. J. Fluid Mech., 60:401-409.Modi, V.J., Wiland, E., Dikshit, A.K. and Yokomizo, T. (1992): On the fluid dynamics of elliptic cylinders. Proc. 2nd Int. Offshore and Polar Engrg. Conf., San Francisco, CA, 14-19 J u n e 1992, 111:595-614.
    • References 35Nikuradse, J. (1933): Stromungsgesetze in rauhen Rohren. Forsch. Arb.Ing.-Wes. No. 361.Norberg, C. and Sunden, B. (1987): Turbulence and Reynolds number effects on the flow and fluid forces on a single cylinder in cross flow. Jour. Fluids and Structures, 1:337-357.Novak, M. and Tanaka, H. (1977): Pressure correlations on a vibrating cylin- der. Proc. 4th Int. Conf. on W i n d Effects on Buildings and Structures, Heathrow, U.K., Ed. by K.J. Eaton. Cambridge Univ. Press, p p . 227-232.Raven, P.W.J., Stuart, R.J., Bray, J.A. and Littlejohns, P.S. (1985): Full-scale dynamic testing of submarine pipeline spans. 17th Annual Offshore Tech- nology Conference, Houston, Texas, May 6-9., paper No. 5005, 3:395-404.Ribeiro, J.L.D. (1992): Fluctuating lift and its spanwise correlation on a circular cylinder in a smooth and in a turbulent flow: a critical review. Jour, of W i n d Engrg. and Indust. Aerodynamics, 40:179-198.Roshko, A. (1961): Experiments on the flow past a circular cylinder at very high Reynolds number. J. Fluid Mech., 10:345-356.Schewe, G. (1983): On the force fluctuations acting on a circular cylinder in cross- flow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech., 133:265-285.Schlichting, G. (1979): Boundary Layer Theory. 7.ed. McGraw-Hill Book Com- pany.Szepessy, S. and Bearman, P.W. (1992): Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech., 234:191-217.Toebes, G.H. (1969): T h e unsteady flow and wake near an oscillating cylinder. Trans. ASME J. Basic Eng., 91:493-502.Williamson, C.H.K. (1988): T h e existence of two stages in the transition to three- dimensionality of a cylinder wake. Phys. Fluids, 31(11):3165-3168.Williamson, C.H.K. (1989): Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds number. J. Fluid Mech., 206:579-627.
    • Chapter 2. Forces on a cylinder in steady current T h e flow around t h e cylinder described in Chapter 1 will exert a resultantforce on the cylinder. There are two contributions to this force, one from thepressure and the other from the friction. T h e in-line component of the mean resultant force due to pressure (thein-line mean pressure force) per unit length of the cylinder is given by _ r2n /•27T Fv = / pcos((j>)r0d<j>, (2.1) Jo(see Fig. 2.1 for t h e definition sketch), while t h a t due t o friction (the in-line meanfriction force) is given by •2TT I TO sin(^)rod</>in which p is the pressure and To is the wall shear stress on the cylinder surface, (2.2)and the overbar denotes time-averaging. T h e total in-line force, the so-called m e a n d r a g , is the sum of these twoforces: FD = FP+Ff (2.3)Fp is termed the f o r m d r a g and Ff the friction drag.
    • Drag and lift SI Figure 2.1 Definition sketch. Regarding the cross-flow component of the mean resultant force, this forcewill be nil due to symmetry in the flow. However, t h e instantaneous cross-flowforce on t h e cylinder, i.e., the instantaneous lift f o r c e , is non-zero and its valuecan be rather large, as will be seen in t h e next sections.2.1 Drag and lift As has been discussed in Chapter 1, the regime of flow around a circularcylinder varies as t h e Reynolds number is changed (Fig. 1.1). Also, effects suchas the surface roughness, the cross-sectional shape, the incoming turbulence, andthe shear in the incoming flow influence the flow. However, except for very smallReynolds numbers (Re ~ 40), there is one feature of the flow which is common toall t h e flow regimes, namely t h e vortex shedding. As a consequence of the vortex-shedding phenomenon, t h e pressure distri-bution around the cylinder undergoes a periodic change as the shedding processprogresses, resulting in a periodic variation in t h e force components on the cylin-der. Fig. 2.2 shows a sequence of flow pictures of the wake together with themeasured pressure distributions and t h e corresponding force components, whichare calculated by integrating the pressure distributions over the cylinder surface(the time span covered in the figure is slightly larger t h a n one period of vortexshedding). Fig. 2.3, on the other hand, depicts the force traces corresponding tothe same experiment as in t h e previous figure. T h e preceding figures show t h e following two important features: first, theforce acting on t h e cylinder in the in-line direction (the drag force) does changeperiodically in time oscillating around mean drag, and secondly, although theincoming flow is completely symmetric with respect to the cylinder axis, thereexists a non-zero force component (with a zero mean, however) on the cylinder
    • Pressure t = 0.84s U- 0.87s 0.90s 0.94s 0.97sFigure 2.2 Time development of pressure distribution and the force components, as the Re = 1.1 X 10 5 , D = 8 cm and U = 1.53 m/s. cp = (p - p 0 ) / ( p U 2 ) . D
    • Drag and lift S9 C C D- L4 2 Vortex- Shedding period Figure 2.3 Drag and lift force traces obtained from the measured pressure distributions in the previous figure. Cp = Fo/^pDU2) and CL = FL/{pDU2). Drescher (1956).in the transverse direction (the lift force), and this, too, varies periodically withtime. In the following paragraphs we will first concentrate our attention on themean drag, then we will focus on the oscillating components of the forces, namelythe oscillating drag force and the oscillating lift force.
    • 40 Chapter 2: Forces on a cylinder in steady current2.2 Mean dragForm drag and friction drag Fig. 2.4 shows the relative contribution to the total mean drag force fromfriction as function of the .Re-number. T h e figure clearly shows t h a t , for therange of Re numbers normally encountered in practice, namely Re ~ 10 4 , t h econtribution of the friction drag to the total drag force is less t h a n 2 - 3 % . So thefriction drag can be omitted in most of the cases, a n d the total mean drag can beassumed to be composed of only one component, namely t h e form drag FD 0.020 o± A u io Thorn ( 1 9 2 9 ) ^ - A 0.010 A • °A 0.005 0.002 0.001 _l 10 10 10 Re 10 Figure 2.4 Relative contribution of the friction force to the total drag for circular cylinder. Achenbach (1968). p cos((f>)rod<f> (2.4) Jo Fig. 2.5a depicts several measured pressure distributions for different valuesof Re, while Fig. 2.5b presents the corresponding wall shear stress distributions.Fig. 2.5a contains also the pressure distribution obtained from t h e potential flowtheory, which is given by P-l -pU2(l-4:Sm2 (2.5)
    • Mean drag ^1 R e = l x 10 2.6x 1 0 5 3.6x 10 6 300 360 < > t (b) ^?f^ 1 1 1 • X" I "^QL Separation point 1 1 1 • W^•*/ kf r"-Mi ""»• i i i , 1 1 tV _ Re=lx l O 5 " " ^ . tjf T * • // 2.6x l O 5 " " ^ N - 3.6x 1 0 6 - ^ ^ ^ -V - 1 1 1 i i i i i i 1 1 1 0 60 120 180 240 300 360Figure 2.5 Pressure distribution and wall shear stress distribution at dif- ferent Re numbers for a smooth cylinder. Achenbach (1968).
    • 42 Chapter 2: Forces on a cylinder in steady current Super Upper Subcritical critical transition Transcritical A , / 150 t^S S (f i 140 A Al u = i i o 0 0 1o o o * 130 _ 0 120 - |° - A 110 - Separation 100 - o point 90 _ % _ .A 80 - 70 i i i i i ml *• i i | | nil 1 1 1 1 1 1 III tab- 4 10 5 10 5 10 Re 10 2 Figure 2.6 Position of the separation point as a function of the Reynolds number for circular cylinder. Achenbach (1968).in which po is t h e hydrostatic pressure. Fig. 2.6 gives t h e position of separationpoints as a function of Re. T h e main characteristic of the measured pressure distributions is t h a t thepressure at the rear side of the cylinder (i.e., in the wake region) is always negative(in contrast to what the potential-flow theory gives). This is due to separation.Fig. 2.5a further indicates t h a t the pressure on the cylinder remains practicallyconstant across the cylinder wake. This is because the flow in t h e wake region isextremely weak as compared to the outer-flow region.D r a g coefficient T h e general expression for t h e drag force is from Eqs. 2.1-2.3 given by /-27T FD= (pcos((f>) +T0sm((f>))rod<f> (2.6) Jo This equation can be written in the following form
    • Mean drag 43in which D = 2r 0 , the cylinder diameter. T h e right-hand-side of the equation is afunction of t h e Re number, since b o t h the pressure term and the wall shear stressterm are functions of the Re number for a smooth cylinder (Fig. 2.5). ThereforeEq. 2.7 may be written in t h e following simple form (2.8) kpDU*CD is called the mean drag coefficient, or in short, the drag coefficient, and is afunction of Re. 0.1 I I U-Ll I U-Ll I l_LLl I l_LLl 1 l l ll I lit | 1_LU | l_LLl I • ~ 0 1 2 3 4 5 6 7 £T 10 10 . 10 , 10 . 10 10 10 .. 10, .10 10 Re Trans- No separation •jy Subcritical critical Lam.„ Transition Fixed ^ r t p /Super pair of s ue" ^ to turbulence CH lHi„ a l critical " P P " r r t c < symme- S n d - in the wake transition trie vor- tices Figure 2.7 Drag coefficient for a smooth circular cylinder as a function of the Reynolds number. Dashed curve: The Oseen-Lamb lam- inar theory (Eq. 5.41). Measurements by Wieselsberger for 40 < Re < 5 X 10 5 and Schewe (1983) for Re > 10 5 . The di- agram minus Schewes data was taken from Schlichting (1979). Fig. 2.7 presents the experimental d a t a together with t h e result of thelaminar theory, illustrating t h e variation of Co with respect to the Re number,while Fig. 2.8 depicts t h e close-up picture of this variation in the most interesting
    • 44 Chapter 2: Forces on a cylinder in steady current *ooo^> ( (a) 1.0 0.5 o 8 °°o 0 O OOOw O >° o *°° I I I I 11 I I I I Ill 4 5 10 Re 10 2 x 10 10 (b) ° „8 0.3 ^4 0.2 0.1 f- 0 ° ° ° o oooo 0 °|Q P P P ° i l ° E I 2 X 10 10 10 Re 10 j i 0.5 (c) 1 3 0 OO _ 0.4 _ """bo o St 0.3 - i 0°°° 0.2 -o Oo 0 o o o o o c o x o o o oo ° 1 o°o° °o oo o 0.1 - 1 1 11 l i 1 1 1 11 1 1 1 1 1 1 1 ll ^ 1 1 1 1 4 5 6 10 Re 10 2 x 10 10 4 » Subcritical / Super- Upper Trans- / critical transition critical Critical Figure 2.8 Drag coefficient, r.m.s. of the lift oscillations and Strouhal number as function of Re for a smooth circular cylinder. Schewe (1983).
    • Mean drag J^5range of Re numbers, namely Re ~ 10*. T h e latter figure also contains informationabout the oscillating lift force and the Strouhal number, which are maintained inthe figure for t h e sake of completeness. T h e lift force d a t a will be discussed laterin the section dealing with the oscillating forces. As seen from Fig. 2.7, Co decreases monotonously with Re until Re reachesthe value of about 300. However, from this Re number onwards, Co assumes apractically constant value, namely 1.2, throughout the subcritical Re range (300 <Re < 3 x 10 5 ). W h e n Re attains the value of 3 X 10 5 , a dramatic change occursin Co, the drag coefficient decreases abruptly and assumes a much lower value,about 0.25, in the neighbouring Re range, the supercritical Re range, 3.5 x 10 5 <Re < 1.5 x 10 6 (Fig. 2.8a). This phenomenon, namely the drastic fall in Co, iscalled the d r a g crisis. T h e drag crisis can best be explained by reference to the pressure diagramsgiven in Fig. 2.5. Note t h a t the friction drag can be disregarded in the analysisbecause it constitutes only a very small fraction of t h e total drag. Re=lxlO Re = 8 . 5 x 1 0 (Subcritical) (Supercritical) Figure 2.9 Pressure distributions. cp = (p — p0)/(^pU2). S denotes the separation points. Achenbach (1968). Two of t h e diagrams, namely the one for Re = 1 x 10 5 (a representative Renumber for subcritical flow regime) and t h a t for Re = 8.5x 10 5 (a representative Renumber for supercritical flow regime) are reproduced in Fig. 2.9. From the figure,
    • 1^6 Chapter 2: Forces on a cylinder in steady currentit is evident t h a t the drag should be smaller in t h e supercritical flow regime t h a nin the subcritical flow regime. Clearly, the key point here is t h a t t h e separationpoint moves from <j>3 = 78° {Re = 1 x 10 5 , the laminar separation) to <j>3 = 140°(Re = 8.5 x 10 5 , the turbulent separation), when the flow regime is changedfrom subcritical to supercritical (Fig. 2.6), resulting in an extremely narrow wakewith substantially smaller negative pressure, which would presumably lead to aconsiderable reduction in the drag. Returning to Figs. 2.7 and 2.8 it is seen t h a t the drag coefficient increasesas the flow regime is changed from supercritical to upper-transition, and then Coattains a constant value of about 0.5, as Re is increased further to transcriticalvalues, namely Re > 4.5 x 10 6 . Again, the change in Co for these higher flowregimes can be explained by reference to the pressure distributions given in Fig.2.5 along with the information about the separation angle given in Fig. 2.6.Effect o f surface r o u g h n e s s In the case of rough cylinders, the mean drag, as in t h e case of smoothcylinders, can be assumed to be composed of only one component, namely the formdrag; indeed, Achenbachs (1971) measurements demonstrate t h a t t h e contributionof the friction drag to the total drag does not exceed 2 - 3 % , thus can be omittedin most of t h e cases (Fig. 2.10). 0.03 0.002 Figure 2.10 Relative contribution of the friction force to the total drag. Effect of cylinder roughness. Achenbach (1971). T h e drag coefficient, Co, now becomes not only a function of Re numberbut also a function of the roughness parameter ks/D Co = Co (Re, ^ ) (2.9)
    • Mean drag ^7 I I I I I I I I I I I 1_| I I L*. 4 5 6 „ 4 10 10 10 Re Figure 2.11 Drag coefficient of a circular cylinder at various surface rough- ness parameters k3/D. Achenbach and Heinecke (1981).in which ks is the Nikuradse equivalent sand roughness. Fig. 2.11 depicts Co plotted as a function of these parameters. T h e wayin which Co varies with Re for a given ka/D is sketched in Fig. 2.12. As seen from t h e figures, the Reynolds-number ranges observed for thesmooth-cylinder case still exist. However, two of the high Re n u m b e r ranges,namely t h e supercritical range and t h e upper transition range seem to mergeinto one single range as the roughness is increased. Furthermore, the followingobservations can be m a d e from the figure: 1) For small Re numbers (i.e., the subcritical Re numbers), Co takes thevalue obtained in the case of smooth cylinders, namely 1.4, irrespective of thecylinder roughness. 2) T h e CD~versus-Re curve shifts towards the lower end of the .Re-numberrange indicated in the figure, as the cylinder roughness is increased. Clearly, thisbehaviour is related to the early transition to turbulence in the b o u n d a r y layerwith increasing roughness. 3) T h e drag crisis, which is characterized by a marked depression in the Cocurve, is not as extensive as it is in the smooth-cylinder case: while Co falls from1.4 to a value of about 0.5 in the case of smooth cylinder, it falls from 1.4 only toa value of about 1.1 in t h e case of rough cylinder with k3/D = 30 x 1 0 - 3 . Thisis directly linked with t h e angular location of t h e separation points. Fig. 2.13compares the latter quantity for cylinders with different roughnesses. It is seent h a t , in the supercritical range, while <j>s is equal to 140° in the case of a smooth
    • J8 Chapter 2: Forces on a cylinder in steady current Super Upper critical transition Subcritical Critical /. Transcritical Re Figure 2.12 General form of CD = Co(Re) curve for a rough cylinder. Smooth Figure 2.13 Circular cylinder. Angular position of boundary-layer separa tion at various roughness parameters. Achenbach (1971).
    • Mean drag 1)9cylinder, it is only 115° for the case of a rough cylinder with ks/D = 4.5 x 10~ 3 .(This is because of t h e relatively weaker m o m e n t u m exchange near the wall inthe case of rough wall due to the larger boundary-layer thickness). Therefore, thepicture given in Fig. 2.9b for t h e smooth-cylinder situation (where (j>s = 140°)will not be the same for the rough cylinder (<j>s = 115°). As a m a t t e r of fact, thepressure-distribution picture for the rough cylinder in consideration (<j>a = 115°)must lie somewhere between the picture given in Fig. 2.9a a n d t h a t given in Fig.2.9b, which implies t h a t the fall in the mean drag due to the drag crisis in this casewill not be as extensive as in the case of a smooth cylinder, as clearly indicated inFig. 2.11. Regarding the transcritical Re numbers in Fig. 2.11, t h e transcritical rangecovers smaller and smaller Re numbers as the roughness is increased. Also, theCD coefficient in t h e transcritical range takes higher and higher values with in-creasing roughness, see Table 2.1. Clearly, this behaviour is closely linked withthe behaviour of the cylinder boundary layer. Finally, Fig. 2.14 gives the dragcoefficient as a function of cylinder roughness for t h e transcritical .Re-numberrange. Table 2.1 Transcritical Re number range for various values of the rela- tive roughness. Data from Fig. 2.11. k3/D Transcritical Cylinder Reynolds roughness number range 0 Re > (3 - 4) x 106 0.75 x i r r 3 Re > 9 x 105 3 x icr- 3 3 Re > 5 x 105 9 x lO Re > 3 x 105 30 x 10~3 Re > (1 - 2) x 105 T h e reader is referred to t h e following work for further details of the effectof the cylinder roughness on the mean drag: Achenbach (1968, 1971) and Giiven,Patel and Farell (1975 and 1977), Giiven, Farell and Patel (1980), Shih, Wang,Coles and Roshko (1993) among others.
    • 50 Chapter 2: Forces on a cylinder in steady current c 1.5 • • _-^-~-" * 1.0 ^ ^ 0.5 ks 3 xl A -5- ° 0 0 1 10 Figure 2.14 Drag coefficient for rough cylinders in the transcritical Re- number range (Table 2.1). Data from Fig. 2.11.2.3 Oscillating drag and lift A cylinder which is exposed to a steady flow experiences oscillating forces ifRe > 40, where t h e wake flow becomes time-dependent (Section 1.1). T h e originof the oscillating forces is t h e vortex shedding. As already discussed in Section1.1, the key point is t h a t t h e pressure distribution around t h e cylinder undergoes aperiodic change as the vortex shedding progresses, resulting in a periodic variationin the force (Figs. 2.2 and 2.3). A close inspection of Fig. 2.2 reveals t h a t theupward lift is associated with the growth of the vortex at the lower edge of thecylinder (t = 0.87 - 0.94 s), while the downward lift is associated with t h a t at theupper edge of t h e cylinder (t = 1.03 - 1.10 s). Also, it is readily seen t h a t b o t hvortices give a temporary increase in the drag. As seen from Fig. 2.3, t h e lift force on the cylinder oscillates at the vortex-shedding frequency, / „ ( = 1/T„), while the drag force oscillates at a frequencywhich is twice t h e vortex-shedding frequency. Fig. 2.3 further indicates t h a t theamplitude of the oscillations is not a constant set of value. As is seen, it variesfrom one period to t h e other. It may even happen t h a t some periods are missed.Nevertheless, t h e magnitude of the oscillations can be characterized by their sta-tistical properties such as the root-mean-square (r.m.s.) value of t h e oscillations.Fig. 2.15 gives the oscillating-force d a t a compiled by Hallam, Heaf and Wootton
    • Oscillating drag and lift 51 CD,Ci w^ cL 0.1 2 (C D ) * a) 0.05 * * * * * * J i i M I i i i i I I I II 10 10 Re b) Range of r e s u l t s for stationary cylinders i i i i i i i i i i i i i i 10" 1 0 7 Re Figure 2.15 R.m.s.-values of drag and lift oscillations. CD = FDj {pDU2) and CL = F[l(pDU2). Hallam et al. (1977).(1977), regarding the magnitude of the oscillations in the force coefficients whereCD and CL are defined by the following equations K = -2PCDBV2 (2.10) (2.11)
    • 52 Chapter 2: Forces on a cylinder in steady currentin which FD is the oscillating part of the drag force FD = FD-FD , (2.12)and F[ is t h e oscillating lift force F[ = FL-FL = FL-0 = FL , (2.13)(CD2) and (CL2) are the r.m.s. values of the oscillations CD and CL,respectively. T h e magnitude of the oscillating forces is a function of Re, which canbe seen very clearly from Fig. 2.8, where CL d a t a from a single set of experimentsare shown along with the Co and the Si-number variations obtained in the samework. It is evident t h a t the r.m.s.-value of CL experiences a dramatic change inthe same way as in the case of Co and St in the critical flow regime, and thenit attains an extremely low value in the supercritical flow regime. This point hasalready been mentioned in Section 1.2.1 in connection with the frequency of vortexshedding with reference to the power spectra of t h e lift oscillations illustrated inFig. 1.10 (cf. Fig. 1.10a and 1.10b, and note the difference in t h e scales of thevertical axes of the two figures). T h e main reason behind this large reduction in ther.m.s.-value of CL is t h a t , in the supercritical flow regime, t h e interaction betweenthe vortices in the wake is considerably weaker, partly because the b o u n d a r y layerseparates at an extremely large angular position (Fig.2.6) meaning t h a t the vorticesare much closer to each other in this flow regime, and partly because the boundary-layer separation is turbulent (Fig. 1.1).2.4 Effect of cross-sectional shape on force coefficients T h e shape of the cross-section has a large influence on the resulting force.A detailed table giving the variation in the force coefficient with various shapesof cross-sections is given in Appendix I. There are two points which need to be elaborated here. One is the Reynoldsnumber dependence in the case of cross-sectional shapes with sharp edges. In thiscase, practically no Reynolds number dependence should be expected since theseparation point is fixed at t h e sharp corners of the cross section. So, no change inforce coefficients is expected with Re number for these cross-sections in contrastto what occurs in t h e case of circular cross-sections. Secondly, non-circular cross-sections may be subject to steady lift at a cer-tain angle of attack. This is due to the asymmetry of t h e flow with respect to theprinciple axis of the cross-sectional area. A similar kind of steady lift has been ob-served even for circular cylinders in t h e critical flow regime (Schewe, 1983) wherethe asymmetry occurs due to t h e one-sided transition to turbulence (Section 1.1). Fig. 2.16 presents the force coefficient regarding this steady lift for differentcross-sections.
    • Effect of incoming turbulence on force coef ficients 53 0 5 10 15 20 25 a(deg) Figure 2.16 Steady lift force coefficients, Re = 33,000 to 66,000. Parkinson and Brooks (1961).2.5 Effect of incoming turbulence on force coefficients T h e turbulence in t h e approaching flow may affect the force coefficients,Cheung and Melbourne (1983), Kwok (1986), and Norberg and Sunden (1987).T h e effect is summarized in Fig. 2.17 based on the d a t a presented in Cheung andMelbourne (1983). T h e dashed lines in t h e figure correspond to t h e case where theturbulence level is very small, and therefore the flow in this case may be consideredsmooth. T h e figures clearly show t h a t the force coefficients are affected quite con-siderably by the incoming turbulence. Increasing the turbulence level from almostsmooth flow (the dashed curves) to larger and larger values acts in t h e same wayas increasing t h e cylinder roughness (cf. Fig. 2.17a and Fig. 2.11). As has beendiscussed in the context of t h e effect of roughness, the increased level of incomingturbulence will directly influence the cylinder boundary layer and hence its sepa-ration. This will obviously lead to changes in the force and therefore in the forcecoefficients.
    • 54 Chapter 2: Forces on a cylinder in steady current Mt. Isa stack full scale data C D =0.6 for I u =6.5%,Re=10 Figure 2.17 Effect of turbulence on the force coefficients. Iu is defined in Eq. 1.8. Cheung and Melbourne (1983).
    • Effect of angle of attack on force coefficients 552.6 Effect of angle of attack on force coefficients W h e n a cylinder is placed at an angle to the flow (Fig. 2.18), forces onthe cylinder may change. Experiments show, however, t h a t in most of the casesthe so-called independence or cross-flow principle is applicable (Hoerner, 1965).Namely, t h e component of t h e force normal to t h e cylinder may be calculated from FN = pCDD U2N (2.14)in which Upi is t h e velocity component normal to the cylinder axis. T h e dragcoefficient in the preceding equation can be taken as t h a t obtained for a cylindernormal t o t h e flow. So, Co is independent of the angle of attack, 8. Figure 2.18 Definition sketch. Angle of attack of flow, 6, is different from 90°. It may be argued t h a t t h e flow sees an elliptical cross-section in t h e case ofan oblique attack, and therefore separation may be delayed, resulting in a value ofCo different from t h a t obtained for a cylinder normal t o t h e flow. Observationsshow, however, t h a t , although t h e approaching flow is at an angle, the streamlinesin the neighbourhood of t h e cylinder are bent in such a way t h a t the actual flowpast t h e cylinder is at an angle of about 8 = 90° (Fig. 2.19). Therefore, theposition of t h e separation point practically does not change, meaning t h a t Coshould be independent of 8. Kozakiewicz, Freds0e and Sumer (1995), based ontheir flow-visualization experiments, give t h e critical value of 8 approximately 35°.For 8 ~ 35°, t h e streamlines do not bend, implying t h a t , for such small values of8, Co is no longer independent of 6, a n d therefore the independence principle willbe violated.
    • 56 Chapter 2: Forces on a cylinder in steady current Figure 2.19 Visualization of flow past a circular cylinder in the case of oblique attack {6 being different from 90°). Kozakiewicz et al. (1995). Regarding the lift, Kozakiewicz et al. (1995) report that the independenceprinciple is valid also for the lift force for the tested range of 6 for their force mea-surements, namely 45° < 6 < 90°. They further report that the vortex sheddingfrequency (obtained from the lift-force spectra) is close to the value calculatedfrom the Strouhal relationship. The lift force power spectrum becomes broader,however, as 6 is decreased. Kozakiewicz et al.s (1995) study covers also the case of a near-bottomcylinder (the pipeline problem) with the gap between the cylinder and the bottombeing 0.1 D in one case and nil in the other. Apparently, the independence principleis valid also for the near-bottom-cylinder situation for the tested range of 6(45° <0 < 90°). Finally, it may be noted that, although, theoretically, the independenceprinciple is justified only in the subcritical range of Re, it has been proved tohold true also in the postcritical flows (Norton, Heideman and Mallard, 1981).However, there is evidence (Bursnall and Loftin, 1951) that for the transcriticalvalues of Re the independence principle may not be applied.
    • Forces on a cylinder near a wall 572.7 Forces on a cylinder near a wall T h e changes in t h e flow caused by the wall proximity is discussed in Section1.2.1; these changes will obviously influence the forces acting on the cylinder. This section will describe the effect of wall proximity on t h e forces on acylinder placed near (or on) a wall. T h e following aspects of t h e problem will beexamined: t h e drag force, t h e lift force, t h e oscillating components of t h e drag andthe lift, and finally the forces on a pipeline placed in/over a scour trench.D r a g force o n a c y l i n d e r n e a r a p l a n e wall Fig. 2.20 depicts t h e pressure distributions around a cylinder placed atthree different distances from a plane wall (Bearman and Zdravkovich, 1978). Fig.2.21, on the other h a n d , presents the experimental d a t a on the drag coefficientfrom the works by Kiya (1968), Roshko, Steinolffron and Chattoorgoon (1975),Zdravkovich (1985) and Jensen, Sumer, Jensen and Freds0e (1990). T h e dragcoefficient is defined in t h e same way as in Eq. 2.8. T h e general trend is t h a t the drag coefficient decreases with decreasing gapratio near the wall. This result is consistent with the pressure distributions givenin Fig. 2.20. T h e differences between the various experiments in t h e figure may be at-tributed t o t h e change in t h e Reynolds number. One characteristic point in the variation of CQ with respect to e/D is t h a t ,as seen from t h e figure, Cp increases in a monotonous manner with increasing e/Dup to a certain value of e/D, and then it remains reasonably constant for furtherincrease in e/D (Fig. 2.22). This behaviour has been linked by Zdravkovich (1985)t o t h e thickness of t h e boundary layer of t h e approaching flow: t h e flat portionof the curve occurs for such large gap ratios t h a t the cylinder is embedded fullyin the potential flow region. At lower gap ratios the cylinder is embedded partlyin the potential flow region and partly in t h e boundary layer of the incomingflow. T h e curves belonging to Zdravkovichs (1985) d a t a in Fig. 2.21 with twodifferent values of S/D, namely S/D = 0.5 a n d S/D = 1 where S = t h e thicknessof the boundary layer in the approaching flow, demonstrates this characteristicbehaviour.Lift force o n a c y l i n d e r n e a r a p l a n e wall T h e m e a n flow around a near-wall cylinder is not symmetric, therefore anon-zero mean lift must exist (in contrast to the case of a free cylinder). Fig. 2.20shows t h a t , while the mean pressure distribution around t h e cylinder is almostsymmetric when e/D = 1, meaning t h a t practically no lift exists, this symmetry
    • 58 Chapter 2: Forces on a cylinder in steady current a)i-l Stagnation 777777 b)^=0.1 TTT777TTJ 7 Stagnation V_V ///////// _^s Figure 2.20 Pressure distributions on a cylinder near a wall as a function of gap ratio e/D. cp = (p — Po)l(pU2) where po s the hydrostatic pressure. Bearman and Zdravkovich (1978).clearly disappears for t h e gap ratios e/D = 0.1 and 0, resulting in a non-zero m e a nlift on the cylinder. This lift, as seen from t h e figure, is directed away from thewall. T h e variation of the lift force with respect to the gap ratio can best bedescribed by reference to the simple case, the shear-free flow situation, depicted inFig. 2.23. In the figure are plotted Freds0e, Sumer, Andersen and Hansens (1985)experimental d a t a , Freds0e a n d Hansens (1987) modified potential-flow solutionand also the potential-flow solution for a wall-mounted cylinder (see, for example,Yamamoto, Nath and Slotta (1974) for the latter). T h e shear-free flow in Freds0eet al.s study was achieved by towing t h e cylinder in still water. T h e CL coefficient
    • Forces on a cylinder near a wall 59 tl^l D y<ua VLV_ Klya (1968), Re = 1 - 4 x 104 1.2H Roshko etal. (1975), Re 2 x 104 Zdravkovlch(1985), Re = 6 - 7 x 104 0.8 S/D=l ditto , Re = 7 - 15x 10 8/D = 0.5 0.4H Jensen et al. (1990), Re = 10 4 0 0.5 1.0 1.5 2.0 e/D Figure 2.21 Drag coefficient for a cylinder near a plane wall, Crj — Fo/i^pUa-D)- In the figure S is the boundary-layer thick- ness of the approaching flow.plotted in the figure is defined by FL = ~PCLDU2 (2.15)where FL is t h e m e a n lift force on t h e cylinder, and t h e positive lift means t h a tit is directed away from t h e wall. T h e figure indicates t h a t while t h e lift is fairly small for gap ratios such ase/D = 0.2 - 0.3, it increases tremendously as the gap ratio is decreased. This isbecause, as mentioned previously, 1) the stagnation point moves to lower and lowerangular positions, as the gap is decreased (Fig. 2.24); also, 2) the suction on thefree-stream side of the cylinder becomes larger and larger with decreasing gaps.T h e combined action of these two effects result in larger and larger lift forces, ast h e cylinder is moved towards the wall. Regarding t h e potential-flow solution plotted in Fig. 2.23, t h e potential flowsolution for a wall-mounted cylinder was given by von Miiller (1929) in closed formas FL - pU2Dir(w2 + 3 ) / 1 8 , which gives a lift force directed away from t h e wallwith a lift coefficient CL = 4.49, as seen in the figure. W h e n the cylinder is placeda small distance away from t h e wall, however, the potential flow solution gives anegative lift, Y a m a m o t o et al. (1974), Freds0e and Hansen (1987). Freds0e and
    • 60 Chapter 2: Forces on a cylinder in steady current a) b) Potential flow Boundary *I TTTTrmmrr, Tint - - layer Cn" e/D Figure 2.22 Schematic variation of drag coefficient with the gap ratio.Hansen modified the potential flow solution by superposing a vortex body aroundthe cylinder onto the existing potential flow such t h a t the velocity at the top andat the b o t t o m of the cylinder becomes equal, in accordance with the experimentalobservation which is referred to in the same study. Freds0e and Hansens modifiedpotential-flow solution, as is seen from Fig. 2.23, agrees quite satisfactorily withthe experimental results. W h e n a shear is introduced in the approaching flow, the variation of thelift force with respect to the gap ratio changes considerably very close to the wall,as seen in Fig. 2.25, where Ci is defined by Eq. 2.15 with U replaced by Ua, theundisturbed flow velocity at the level of the cylinder axis. T h e shear-flow d a t aplotted in this figure were obtained in an experiment conducted at practically thesame Reynolds number, employing the same test cylinder as in Fig. 2.23. T h eonly difference between the two tests is t h a t in the shear-free flow experiments thecylinder was towed in still water, while in the shear-flow experiments the cylinderwas kept stationary and subject to the boundary-layer flow established in an openchannel with a smooth b o t t o m . Clearly, the difference observed in Fig. 2.25 in the Cr, versus e/D behaviouris due to the shear in the approaching flow. T h e lift undergoes a substantial dropfor very small gap ratios. Freds0e and Hansen (1987) links this drop to t h e changein the stagnation pressure in the following way: First they show t h a t t h e stagnationpoint does not move significantly by t h e introduction of the shear. So the directionof pressure force is much t h e same in b o t h cases. T h e major difference is t h a t t h estagnation pressure is reduced considerably with the introduction of the shear,
    • Forces on a cylinder near a wall 61 u FL 4.8 4.6 11111 n nun dh i Vjj/ _ in i II i in i D [] e 4.4 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 e/D Figure 2.23 Lift force for a cylinder in a shear-free flow Ci = Fr,/(hpU2D). 4 4 Circles: Experiments, 10 < Re < 3 x 10 (Freds0e et al., 1985). Solid curve: Freds0e and Hansens (1987) modified potential- flow solution. Square: Potential-flow solution (see for example Yamamoto et al., 1974).as sketched in Fig. 2.26; while the stagnation pressure in the shear-free flow,implementing t h e Bernoulli equation and taking t h e far-field pressure, is equal to V=-2PU (2.16)the same quantity in the case of shear flow, to a first approximation, is pul (2.17)where Us is the far-field flow velocity associated with the stagnation streamline. Clearly, the pressure in Eq. 2.17 is much smaller t h a n t h a t in Eq. 2.16 (Fig.2.26). This reduction in the stagnation pressure, while keeping the direction of
    • 62 Chapter 2: Forces on a cylinder in steady current Figure 2.24 Sketches showing the changes in the stagnation point and the pressure distribution, as the cylinder is moved towards the wall: The stagnation point moves to lower and lower angular posi- tions, and the suction on the free-stream side of the cylinder becomes larger and larger than that on the wall side. 0.8 QI° i rim minion 0.6 0.4 minimi f 0.2 0 0 0.1 0.2 0.3 0.4 e/D Figure 2.25 Comparison of Ci in shear-free and shear flows, 10 3 < Re < 3 X 10 . The boundary-layer thickness to diameter ratio 8/D = 5. In the shear flow case Ci is defined by Fi = ^pCiDU^ where Ua is the undisturbed velocity at the axis of the cylinder. Freds0e et al. (1985).
    • Forces on a cylinder near a wall 63 u / iPU lllllh iiiiinim<JJJ>IIII linniiiiiiiii Figure 2.26 Comparison of shear-free and shear flows. Stagnation pressure decreases considerably in the shear-flow case.pressure forces unchanged, presumably causes t h e lift to be reduced substantiallyin t h e case of shear flow. W h e n t h e cylinder is moved extremely close to t h e wall, however, more a n dmore fluid will be diverted to pass over the cylinder, which will lead to larger andlarger suction pressure on the free-stream side of the cylinder. Indeed, when thecylinder is sitting on t h e wall, the suction pressure on t h e cylinder surface will bethe largest (Fig. 2.20c). This effect may restore t h e lift force in the shear-flowcase for very small gap values, as is implied by Fig. 2.25. Fig. 2.27 presents d a t a regarding the lift on a cylinder in a shear flowobtained at different Reynolds numbers.O s c i l l a t i n g d r a g a n d lift o n a c y l i n d e r n e a r a p l a n e wall T h e vortex-induced, oscillating lift a n d drag will cease to exist in the casewhen the gap ratio is smaller t h a n about 0.3, simply because t h e vortex sheddingis suppressed for these gap ratios (Section 1.2.1). Although the shedding exists for gap ratios larger t h a n 0.3, it will, however,be influenced by the close proximity of the wall when e/D is not very large.Therefore t h e oscillating forces will be affected, too, by t h e close proximity of t h ewall. Fig. 2.28 illustrates this influence regarding the r.m.s.-value of the oscillatinglift force. T h e figure shows t h a t t h e oscillating lift becomes weaker and weaker, asthe gap ratio is decreased. Note t h a t t h e CL coefficient here is defined in t h e sameway as in Eq. 2.11 provided t h a t U is replaced by the velocity Ua, t h e undisturbedflow velocity at the level of the cylinder axis. Finally, Fig. 2.29 compares the vortex-shedding induced oscillating lift withthe mean lift caused by t h e wall proximity. T h e CL coefficient plotted in the figurerepresenting t h e vortex-induced oscillating lift is the lift coefficient associated withthe m a x i m u m value of the oscillating lift force. As is seen from t h e figure, thewall-induced lift and t h e vortex-induced lift appear t o be in t h e same order of
    • 64 Chapter 2: Forces on a cylinder in steady current 1111111111111111II1111 Freds0e etal.(1985); 10 4 <Re<3x 104 Thomschke(1971); Re = 9.2x 10" Tliomschke(1971); Re = 2.1x10 5 0.8 • Jones(1971); Re 5 10 5 0.6 0.4 0.2 0.5 e / D Figure 2.27 Lift force on a near-wall cylinder in a shear flow. CL=FLI{PUID).magnitude in t h e neighbourhood of e/D = 0.3. W i t h decreasing values of e/D,however, the wall-induced lift increases quite substantially. T h e figure furtherindicates t h a t , with e/D larger t h a n 0.3 up to 0.4 - 0.5, t h e two effects, namelythe wall-induced steady lift force and the vortex-induced oscillating lift force, maybe present concurrently, meaning t h a t , while the cylinder undergoes a steady lift,it will also be subject to an oscillating lift force induced by vortex shedding.Forces o n a p i p e l i n e i n / o v e r a s c o u r t r e n c h As mentioned in Section 1.2.1, when a pipeline is placed on an erodible bed,scour may occur below the pipe due to flow action, leading to suspended spansof the pipeline. Jensen et al. (1990) investigated the flow around and forces ona pipeline (placed initially on a flat bed) at five characteristic stages of the scourprocess. T h e results regarding the flow description have been given in Section 1.2.1under the heading "Effect of wall proximity" (Figs. 1.24 and 1.25). Fig. 2.30 givesthe force coefficients obtained in the same study. T h e force coefficients are defined,based on the undisturbed velocity at the axis of the pipe. As mentioned in the flow
    • Forces on a cylinder near a wall 65 7 11111 :oi! 77777777 mi mini 2i (CL)24 (C L 2 ) 2 a s e / D -» °° 0.3 — / 0.2 0.1 0 0.2 0.4 0.6 0.8 1.0 e/D Figure 2.28 ii.m.s.-value of oscillating lift coefficient. CL = F[/(^pU^D). Re = 10 4 . Circles: Jensen et al. (1990). Asymptotic value for e/D = oo from Schewe (1983).description, each profile corresponds to a particular instant in the course of thescourjjrocess from which the profiles are taken. It is interesting to note that CQa n d CL reach their equilibrium values at rather early stages of t h e scour process.It is also interesting to observe t h a t the pipe experiences a negative lift force assoon as the tunnel erosion (Stage II) comes into action. It is seen t h a t this liftforce remains negative throughout the scour process. T h e negative lift in Stage II can be a t t r i b u t e d to the strong suction belowand behind the cylinder^ caused by the gap flow, which is also the cause of therelatively high value of CD obtained for Stage II. As for Stage V, t h e negative liftcan be explained by t h e position of t h e stagnation point a n d t h e angle of attackof the approaching flow. This angle can in Fig. 1.24 be found to be around 10-15degrees, which fits well with the angle of the resultant force vector with respect tothe horizontal. T h e phenomenon, namely the " p r e m a t u r e " vortex shedding, which causest h e high Strouhal numbers in the initial stages of the scour process (Stages IIIand IV in Fig. 1.25), is also the main cause of t h e variation in the mean doubleamplitude of t h e fluctuating lift force: t h e larger t h e strength of t h e vortices shed,the larger the fluctuating lift force. Since the vortices shed from the pipe becomestronger and stronger as t h e scour progresses, the fluctuating lift force should
    • 66 Chapter 2: Forces on a cylinder in steady current u —, —» e mi riiiiiiiiuii Vortex s h e d d i n g Vortex i n d u c e d oscillating lift,C L e/D Figure 2.29 Force coefficients of the mean lift force (CL) and the oscillating lift force (CL) on cylinder as a function of the gap ratio. The coefficient CL is based on the amplitude of the oscillating lift force.correspondingly increase, as indicated by Fig. 2.30c. Figure 2.31 compares the results presented in Fig. 2.30 with those obtainedwith a plane bed in t h e same study. T h e plane-bed counterpart of each scourprofile is selected on the basis of equal non-dimensional clearance between thepipe and the bed (i.e., equal to e/D, see Fig. 2.31). As seen from the figure, CD is n ° t affected much, whether the bed is a planebed or a scoured one. As for the mean lift coefficient CL, the difference betweena plane bed and a scoured bed is t h a t t h e pipe experiences a negative lift forcein the case of a scoured bed, while it experiences a positive one when the bed isplane (Fig. 2.25). As for t h e fluctuating lift force CL there is practically no difference betweena plane bed and a scoured bed for large values of e/D. However, this is not thecase for small values of e/D, where t h e effect of upstream slope of the dune behind
    • Forces on a cylinder near a wall 67 n 77 a) 1.0 0.5 t =0 time (mln) b) 0.5 t =0 0 -0.5 "1 i i i 11 11 1— time • • •• * (min) 0 2 x CL 1.0 0.5 time (min) 1 10 100 Figure 2.30 Time development of force during the scour process below a pipeline, (a) mean drag coefficient; (b) mean lift coefficient; (c) mean double amplitude of the fluctuating lift force. Jensen et al. (1990).the pipeline is felt very strongly in the vortex-shedding process, as explained inthe flow description in conjunction with Fig. 1.25 in Section 1.2.1. Stansby and Starr (1992) report the results of measurements of drag on apipe undergoing a gradual sinking, as t h e scour process progresses in a live, sandbed. According to Stansby and Starr, t h e drag coefficient is reduced from CD = 1for a pipe sitting on the bed to CD = 0.3-0.4 when the pipe sinked in the sand toa level of about e/D = — 0.6. This is obviously due to the fact t h a t the pipe isprotected against the flow, as it is buried in the sand bed.
    • 68 Chapter 2: Forces on a cylinder in steady current a) 1.0 0.5 time (min) b) 0.5 time -0.5 (mln) 0 2x CL 1.0 " 0.5 .* time -*- (min) 1 10 100 Figure 2.31 Comparison of forces between the cases of a cylinder over a scoured and a plane bed. (a) mean drag coefficient; (b) mean lift coefficient; (c) mean double amplitude of the fluctuating lift force. Jensen et al. (1990). W h e n the pipelines are placed in a trench hole, the forces are reduced con-siderably (Fig. 2.32). As seen, both the drag and the lift are reduced by a factor5-10, depending on the position of the pipe in t h e trench hole. This is becausethe pipe is protected against the main body of the flow by the trench (shelteringeffect). Jensen and Mogensen (1982) report t h a t in the case of a trench hole thesame size as t h a t in Fig. 2.32 but with a much steeper slope (namely 1:1), thereduction in the forces is even much larger.
    • Forces on a cylinder near a wall 69 Trench 1:5 1 i y 2.5 D | 1 5D L i IV IV0.20 - III X 0.20 - III ^0.16 -0.12 - II V " " " • " • - • 0.16 - 0.12 - II XX ^0.08 - 0.08 - x ~ 1 1— 1 - —• — 1 — • R e x 10 Rex 10 Figure 2.32 Relative drag and lift forces on a pipeline placed in a trench for several positions (Positions II, III and IV). FDI and FLI are the corresponding forces on the same pipeline sitting on a flat bed (Position I). Jensen and Mogensen (1982).
    • 70 Chapter 2: Forces on a cylinder in steady currentREFERENCESAchenbach, E. (1968): Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to Re = 5 x 10 s . J. Fluid Mech., 34(4):625- 639.Achenbach, E. (1971): Influence of surface roughness on the cross-flow around a circular cylinder. J. Fluid Meek, 46:321-335.Achenbach, E. and Heinecke E. (1981): On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 X 103 to 5 x 10 6 . J. Fluid Mech., 109:239-251.Bearman, P.W. and Zdravkovich, M.M. (1978): Flow around a circular cylinder near a plane boundary. J. Fluid Mech., 89(l):33-48.Bursnall, W.J. and Loftin, L.K. (1951): Experimental Investigation of the Pres- sure Distribution about a Yawed Circular Cylinder in the Critical Reynolds Number Range. NACA, Technical Note 2463.Cheung, J.C.K. and Melbourne, W.H. (1983): Turbulence effects on some aerody- namic parameters of a circular cylinder at supercritical Reynolds numbers. J. of Wind Engineering and Industrial Aerodynamics, 14:399-410.Drescher, H. (1956): Messung der auf querangestromte Zylinder ausgeiibten zeitlich veranderten Driicke. Z. f. Flugwiss, 4(112):17-21.Freds0e, J. and Hansen, E.A. (1987): Lift forces on pipelines in steady flow. J. Waterway, Port, Coastal and Ocean Engineering, ASCE, 113(2):139-155.Freds0e, J., Sumer, B.M., Andersen, J. and Hansen, E.A. (1985): Transverse vibrations of a cylinder very close to a plane wall. Proc. 4th Symposium on Offshore Mechanics and Arctic Engineering, OMAE, Dallas, TX, 1:601-609. Also, Trans, of the ASME, J. Offshore Mechanics and Arctic Engineering, 109:52-60.Giiven, O., Patel, V.C. and Farell, C. (1975): Surface roughness effects on the mean flow past circular cylinders. Iowa Inst. Hydraulic Res., Rep. No. 175.Giiven, O., Patel, V.C. and Farell, C. (1977): A model for high-Reynolds-number flow past rough-walled circular cylinders. Trans. ASME, J. Fluids Engrg., 99:486-494.
    • References 71Giiven, O., Farell, C. and Patel, V.C. (1980): Surface-roughness effects on the mean flow past circular cylinders. J. Fluid Mech., 98(4):673-701.Hallam, M.G., Heaf, N.J. and Wootton, L.R. (1977): Dynamics of Marine Struc- tures. CIRIA Underwater Engineering Group, Report UR8, Atkins Re- search and Development, London, U.K.Hoerner, S.F. (1965): Fluid-Dynamic Drag. Practical Information on Aerody- namic Drag a n d Hydrodynamic Resistance. Published by t h e Author. Ob- tainable from ISVA.Jensen, R. and Mogensen, B. (1982): Hydrodynamic forces on pipelines placed in a trench under steady current conditions. Progress Report No. 57, Inst, of Hydrodynamics and Hydraulic Engineering, ISVA, Techn. Univ. Denmark, p p . 43-50.Jensen, B.L., Sumer, B.M., Jensen, H.R. and Freds0e, J. (1990): Flow around and forces on a pipeline near a scoured bed in steady current. Trans, of the ASME, J. of Offshore Mech. and Arctic Engrg., 112:206-213.Jones, W . T . (1971): Forces on submarine pipelines from steady currents. Paper presented at the Petroleum Mechanical Engineering with Underwater Tech- nology Conf., Sept. 19-23, 1971, Houston, Texas, Underwater Technology Div., ASME.Kiya, M. (1968): Study on the turbulent shear flow past a circular cylinder. Bul- letin Faculty of Engrg., Hokkaido University, 50:1-100.Kozakiewicz, A., Freds0e, J. and Sumer, B.M. (1995): Forces on pipelines in oblique attack. Steady current and waves. Proc. 5th Int. Offshore and Polar Engineering Conf., T h e Hague, Netherlands, J u n e 11-16, 1995, 11:174- 183.Kwok, K.C.S. (1986): Turbulence effect on flow around circular cylinder. J. En- gineering Mechanics, ASCE, 112(11):1181-1197.Miiller, W. von (1929): Systeme von Doppelquellen in der ebenen Stromung, ins- besondere die Stromung u m zwei Kreiszylinder. Zeitschrift fur angewandte M a t h e m a t i k und Mechanik, 9(3):200-213.Norberg, C. and Sunden, B. (1987): Turbulence and Reynolds number effects on t h e flow and fluid forces on a single cylinder in cross flow. Jour. Fluids and Structures, 1:337-357.
    • 72 Chapter 2: Forces on a cylinder in steady currentNorton, D.J., Heideman, J.C. and Mallard, W . W . (1981): W i n d tests of inclined circular cylinders. Proc. 13th Annual O T C in Houston, T X , May 4-7, O T C 4122, p p . 67-70.Parkinson, G.V. and Brooks, N.P.H. (1961): On the aeroelastic instability of bluff cylinders. J. Appl. Mech., 28:252-258.Roshko, A., Steinolffron, A. and Chattoorgoon, V. (1975): Flow forces on a cylin- der near a wall or near another cylinder. Proc. 2nd US Conf. W i n d Engrg. Research, Fort Collins, Co., Paper IV-15.Schewe, G. (1983): On the force fluctuations acting on a circular cylinder in cross- flow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech., 133:265-285.Schlichting, G. (1979): Boundary Layer Theory. 7.ed. McGraw-Hill Book Com- pany.Shih, W.C.L., Wang, C , Coles, D. and Roshko, A. (1993): Experiments on flow past rough circular cylinders at large Reynolds numbers. J. W i n d Engrg. and Industrial Aerodynamics, 49:351-368.Stansby, P.K. and Starr, P. (1992): On a horizontal cylinder resting on a sand bed under waves and current. Int. J. Offshore and Polar Engrg., 2(4):262-266.Thorn, A. (1929): An investigation of fluid flow in two dimensions. Aero. Res. Counc. London, R. and M. No. 1194, p p . 166-183.Thomschke, H. (1971): Experimentelle Untersuchung der stationaren Umstromung von Kugel und Zylinder in Wandnahe. Fakultat fur Maschinenbau der Uni- versitat Karlsruhe, Karlsruhe, West Germany.Yamamoto, T., Nath, J.H. and Slotta, L.S. (1974): Wave forces on cylinders near plane boundary. J. Waterway, Port, Coastal Ocean Div., ASCE, 100(4):345- 360.Zdravkovich, M.M. (1985): Forces on a circular cylinder near a plane wall. Applied Ocean Research, 7:197-201.
    • Chapter 3. Flow around a cylinder in os- cillatory flows As shown in Chapter 1, the hydrodynamic quantities describing the flowaround a smooth, circular cylinder in steady currents depend on the Reynoldsnumber. In the case where the cylinder is exposed to an oscillatory flow an ad-ditional p a r a m e t e r - the so-called Keulegan-Carpenter number - appears. T h eKeulegan-Carpenter number - the KC number - is defined by KC=Um±w_ TJ T (31)in which Um is the m a x i m u m velocity and Tw is the period of the oscillatory flow.If the flow is sinusoidal with the velocity given by U = Um sin(u;<) (3.2)then the m a x i m u m velocity will be 2-7ra Um = au = — (3.3)where a is the amplitude of the motion. For the sinusoidal case the KC numberwill therefore be identical to K C ^ (3.4)T h e quantity 10 in Eq. 3.2 is the angular frequency of the motion
    • 74 Chapter 3: Flow around a cylinder in oscillatory flows "> = 2*/„, = ^~ (3.5) -* win which fw is the frequency. T h e physical meaning of the KC number can probably be best explainedby reference to Eq. 3.4. T h e numerator on the right-hand-side of the equation isproportional to the stroke of the motion, namely 2a, while the denominator, thediameter of the cylinder D, represents t h e width of the cylinder (Fig. 3.1). SmallKC numbers therefore mean t h a t t h e orbital motion of t h e water particles is smallrelative to the total width of the cylinder. W h e n KC is very small, separationbehind the cylinder may not even occur. Figure 3.1 Definition sketch. Large KC numbers, on the other hand, mean that the water particles travelquite large distances relative to the total width of the cylinder, resulting in sepa-ration and probably vortex shedding. For very large KC numbers (KC — oo), >we may expect t h a t the flow for each half period of t h e motion resembles thatexperienced in a steady current.3.1 Flow regimes as a function of KC number Fig. 3.2 summarizes the changes t h a t occur in the flow as the Keulegan-Carpenter number is increased from zero. T h e picture presented in t h e figure isfor Re = 10 3 in which Re is defined as Re = ^ (3.6) v(As Re is changed, the flow regimes shown in Fig. 3.2 may also change, accom-panied by possible changes at the upper and lower limits of the indicated KC
    • Flow regimes as a function of KC number 75a) No separation. Creeping (laminar) flow. KC < 1.1b) o Separation with Honji vortices. —>—1 S e e Figs. 3 . 3 and 1.1 < KC < 1.6 3.40 A pair of symmetric ^ ^ V _ vortices 1 6 < KC < 2 1 LJ^d) A A pair of symmetric vortices. 2.1 < KC < 4 Turbulence over Ljp^ t h e cylinder surface (A). ns^e) A pair of asymmetric vortices 4 < KC < 7fl cr? Vortex shedding 7 < KC Shedding regimes Figure 3.2 Regimes of flow around a smooth, circular cylinder in oscillatory flow. Re = 10 3 . Source for KC < 4 is Sarpkaya (1986a) and for KC > 4 Williamson (1985). Limits of the KC intervals may change as a function of Re (see Figs. 3.15 and 3.16).
    • 76 Chapter S: Flow around a cylinder in oscillatory flowsranges. We shall concentrate our attention first on the KC dependence, however.T h e influence of Re will be discussed in Section 3.3). As seen from Fig. 3.2, for very small values of KC, no separation occurs, asexpected. T h e separation first appears when KC is increased to 1.1; this occursin the form of the so-called H o n j i i n s t a b i l i t y (Figs. 3.3 and 3.4). W h e n thisKC number is reached, the purely two-dimensional flow over the cylinder surfacebreaks into a three-dimensional flow p a t t e r n where equally-spaced, regular streaksare formed over t h e cylinder surface, as sketched in Fig. 3.3. These streaks can bemade visible by flow-visualization techniques. Observations show t h a t the markedfluid particles, which were originally on the surface of t h e cylinder, would alwaysend up in these narrow, streaky flow zones. T h e observations also show t h a tthese streaks eventually are subject to separation in every half period prior tothe flow reversal, each separated streak being in the form of a mushroom-shapevortex (Figs.3.3 and 3.4). This phenomenon was first reported by Honji (1981) andlater by Sarpkaya (1986a). Subsequently, Hall (1984) carried out a linear stabilityanalysis and showed t h a t t h e oscillatory viscous flow becomes unstable t o axiallyperiodic vortices (i.e. Vortices B in Fig. 3.3) above a critical KC number for agiven Re, validating the experimentally observed flow instability. Figure 3.3 Honji streaks, which are subject to separation in the form of mushroom-shape vortices; see the photograph in Fig. 3.4, viewed in a-a, for the separated, mushroom-shape vortices.
    • Flow regimes as a function of KC number 77 Figure 3.4 Separated mushroom-shape vortices (A) viewed in a-a indicated in Fig. 3.3. Oscillatory flow is in the direction perpendicular to this page. From Honji (1981) with permission - see Credits. T h e flow regime where separation takes place in the form of Honji instabilityoccurs in a narrow KC interval, namely 1.1 < KC < 1.6 (Fig. 3.2b). W i t h afurther increase of KC number, however, separation begins to occur in the formof a pair of symmetric, ordinary, attached vortices as indicated in Fig. 3.2c and d.This regime covers the KC range 1.6 < KC < 4 with the subrange 2.1 < KC < 4where turbulence is observed over the cylinder (Sarpkaya, 1986a). It must beremembered t h a t the limits for the indicated KC ranges in t h e figure are thosefor Re = 1000. W h e n KC is increased even further, the symmetry between the two attachedvortices breaks down. (The vortices are still attached, and no shedding occurs,however). This regime prevails over t h e KC range 4 < KC < 7 (Fig. 3.2e). T h esignificance of this regime is t h a t the lift force is no longer nil, and this is due tothe asymmetry in the formation of the attached vortices. Fig. 3.5 illustrates the time evolution of vortex motions as the flow pro-gresses for t h e regimes where separation occurs in the form of a pair of symmetri-cally attached vortices, namely for the KC range 1.6 < KC < 4 (Fig. 3.2c and d).T h e arrows in t h e figure refer to the cylinder motion in an otherwise still fluid. Asseen from t h e figure, the vortices which form behind the cylinder (Vortex M) arewashed over the cylinder by the end of the previous half period and form a pairof vortex pairs with the newly formed vortices (Vortex N) which would eventuallymove away from the cylinder due to the self-induced velocity fields of t h e vortexpairs.
    • 78 Chapter 3: Flow around a cylinder in oscillatory flows 1 2 3 t ku) M M M 4 5 t 1 S ML £- Vortex pair M CM Figure 3.5 1.6 < KC < 4. Re = 10 3 . Evolution of vortex motions for the regime with a pair of separation vortices (Fig. 3.2c-d). Arrows refer to cylinder motion. The vortices are viewed from a fixed camera. Williamson (1985). Returning to Fig. 3.2, with a further increase of t h e Keulegan-Carpenternumber, we come to the so-called v o r t e x - s h e d d i n g r e g i m e s (KC > 7) (Fig.3.2f). T h e following section will focus on these flow regimes.3.2 Vortex-shedding regimes T h e vortex-shedding regimes have been investigated extensively by, amongothers, Bearman, G r a h a m and Singh (1979), Singh (1979), Grass and K e m p(1979), B e a r m a n and G r a h a m (1979), Bearman, G r a h a m , Naylor and Obasaju(1981) and more recently by Williamson (1985). These works have shed consid-erable light on the understanding of the complex behaviour of vortex motions invarious regimes. Based on the previous research and his own work, Williamson(1985) has described the vortex trajectory patterns in quite a systematic manner.T h e following description is mainly based on Williamson (1985). In the vortex-shedding regimes the vortex shedding occurs during the courseof each half period of t h e oscillatory motion. There are several such regimes, eachof which has a different vortex flow p a t t e r n , observed for different ranges of the
    • Vortex-shedding regimes 7<KC<15. Single-Pair regime 1 2 3 f M M G N 5 6 4 0M t1 M 1 M /^ 0 -G p N 3 1 9/i? 2 t a,R t &k>N R o «.P t q 1 5 ?V° 6 P Q/(TQ s o Q i 1 iure 3.6 a) 7 < KC < 13. The arrows refer to cylinder motion, but the vortices are viewed from a reference frame which moves with the cylinder. The wake consists of a series of vortices convecting out to one side of the cylinder in the form of a street (the transverse vortex street). b) 13 < KC < 15. The wake consists of a series of pairs con- vecting away each cycle at around 45° to the flow oscillation direction, and on one side of the cylinder only. Both in (a) and in (b), there is always one pair of vortices which convect away from the cylinder. Williamson (1985).
    • 80 Chapter 3: Flow around a cylinder in oscillatory flowsKC number. These KC ranges are 7 < KC < 15, 15 < KC < 24, 24 < KC < 32,32 < KC < 40, etc.7 < K C < 15 ( s i n g l e - p a i r r e g i m e ) Fig. 3.6 illustrates the time development of vortex motions in this regime.T h e major portion of the KC range, namely 7 < KC < 13 (Fig. 3.6a), is knownas the transverse-vortex-street regime. Figure 3.7 Transverse-street wake for KC = 12. In this photograph the cylinder is moving up, and is near the end of a half cycle. Due to the induced velocities of the main vortices, one of which is shed in each half cycle, the trail of vortices convects away at around 90° to the oscillation direction in the form of a street. In this case the street travels to the right. Williamson (1985) with permission - see Credits. Fig. 3.6a, Frame 1, indicates t h a t Vortex N has just been shed and thereis a growing vortex (Vortex M) at the other side of the cylinder. W h e n the flowreverses (Fig. 3.6a, Frame 2), b o t h vortices are washed over the cylinder. Asthe half period progresses, Vortex M itself is shed and, being a free vortex, itforms a vortex pair with Vortex N (Fig. 3.6a, Frame 4). T h e vortex pair M +
    • Vortex-shedding regimes 81N will then move away from the cylinder under its self-induced velocity field. Asimplied in the preceding, the concept "pairing" here means t h a t two vortices, ofopposite sign, come together and each is convected by the velocity field of theother. It is evident from the figure t h a t there will be one vortex pair convectingaway from t h e cylinder at t h e end of each full period. This would apparently leadto a t r a n s v e r s e v o r t e x s t r e e t (i.e., a vortex street in the direction perpendicularto the flow direction), as depicted in Fig. 3.7: in this figure, the vortex street isformed at the lower side of the cylinder. Observations show, however, t h a t thevortex street changes sides occasionally. T h e position of the vortex street relativeto the cylinder may be important from the point of view of the lift force acting onthe cylinder. Due to the asymmetry, a non-zero mean lift must exist in this flowregime. W h e n the vortex street changes side, then the direction of this lift forcewill change correspondingly. 1 5 < KC < 2 4 . D o u b l e - P a i r r e g i m e 3 t t a M v 4 6 R 3fc 5 P R 3PB 6 9 1 Figure 3.8 15 < KC < 24. The arrows refer to cylinder motion, but the vortices are viewed from a reference frame which moves with the cylinder. The wake is the result of two vortices being shed in each half cycle. Two trails of vortex pairs convect away from the cylinder in opposite directions and from opposite sides of the cylinder (for example vortices N + M and P + Q). Williamson (1985). Regarding the second portion of the KC range, namely 13 < KC < 15(Fig. 3.6b), the p a t t e r n of vortex motions changes somewhat in this range of KCnumber; the pairs now convect away at around 45° to the flow direction, and this
    • 82 Chapter 3: Flow around a cylinder in oscillatory flowsoccurs at one side only. From both Fig. 3.6a and Fig. 3.6b it is seen t h a t there is always one vortexpair convecting away from t h e cylinder in one period of t h e motion; this is, forexample in Fig. 3.6a, the pair M + N, while in Fig. 3.6b it is N + R in the firstperiod and P + Q in t h e following period. For this reason Williamson (1985) callsthis regime (7 < KC < 15) the single-pair regime.15 < K C < 2 4 (double-pair regime) and further K C regimes Fig. 3.8 gives the time development of vortex motions in t h e case when15 < KC < 24, while Fig. 3.9 gives t h a t in the case when 24 < KC < 32. T h edetailed descriptions are given in the figure captions. However, it is readily seent h a t there are two vortex pairs convecting away from t h e cylinder in t h e formercase, while there are three vortex pairs convecting away from t h e cylinder in thelatter case. 24< KC< 3 2 . Three-Pairs regime 2 3 t 1 N t N P • 9 ( 4 5 6 t f 5 QT j i gf mi * PrQ Tit? R p^f9 J Figure 3.9 24 < KC < 32. The arrows refer to cylinder motion, but the vortices are viewed from a reference frame which moves with the cylinder. The wake is the result of three vortices being shed in a half cycle, and comprises three vortex pairings in a cycle (for example vortices P + Q, N + R and S + T). Williamson (1985). For further KC regimes, the number of vortex pairs will be increased byone each time t h e KC regime is changed to a higher one; the number of vortex
    • Vortex-shedding regimes 83pairs which are convecting away from t h e cylinder will be four in the case when32 < KC < 40 and five in t h e case when 40 < KC < 48 and so on. This meanst h a t there will be two more vortex sheddings in one full period each time the KCrange is changed to a higher regime. This result is a direct consequence of theStrouhal law in oscillatory flows, as shown in Example 3.1.Example 3.1: Consider the oscillatory flow given in Fig. 3.10a. Its KC number is 2a) * C = » (3.7) ~D Now, suppose t h a t we increase t h e KC number by A(KC) = 8 so t h a t thenumber of vortices shed for one full period is increased by 2, or for one half periodby 1 (Fig. 3.10b). In this new situation, KC number will be KC + A(KC) = ^ ± ^ (3.!in which £ is t h e increase in the double-amplitude of the motion. Since A(KC) is8, t h e length £ from Eqs. 3.7 a n d 3.8 will then be e=-D (3.9) •K Given the fact t h a t the increase in t h e number of vortex sheddings in onehalf period is 1, t h e size of £ should then be just enough to accommodate onecomplete vortex shedding (Fig. 3.10b). In other words, t h e time period duringwhich the cylinder travels over t h e length £ should be identical to half of thevortex-shedding period, (1/2)T„: (i T 0 U (3.10)in which U is the average velocity of t h e cylinder during this travel. From Eqs.3.9 and 3.10, the frequency of the vortex shedding, fv = ^r (3.11)will then be: f -£ = = ™ (3-12)
    • 84 Chapter 3: Flow around a cylinder in oscillatory flows 2 a (stroke) shedding r a) o 0 o O shedding one m o r e shedding b) o °0 £ °0 1- 2a Figure 3.10 Definition sketch. In (b): KC number is increased such that the new KC number is in the next, higher KC regime. As is seen, this is nothing but the Strouhal law with the normalized fre-quency being 0.20. So, as a conclusion it may be stated t h a t the observed increasein the number of vortices shed, namely 2 in one full period when KC range ischanged to a higher regime, is a direct consequence of the familiar Strouhal law.V o r t e x - s h e d d i n g f r e q u e n c y a n d lift f r e q u e n c y In contrast to steady currents, the concept "frequency of vortex shedding"is not quite straightforward in oscillatory flows, particularly for lower KC regimessuch as the single-pair regime and the double-pair regime. This is mainly due to thepresence of flow reversals. T h e subject can probably be best explained by referenceto Figs. 3.11 and 3.12. These figures depict time series of the lift force acting ona cylinder and the corresponding motion of vortices, which are reproduced fromFigs. 3.6a a n d 3.8. ( T h e force time series have been obtained simultaneously withthe flow visualizations of vortex motions so t h a t a direct relation between the liftvariation and the motion of vortices could be established, Williamson (1985)). In Fig. 3.11, each negative peak (marked A and C) is caused by the growthand shedding of a vortex (such as N, Frame 1, and M, Frame 4) during each halfperiod, in exactly the same fashion as in steady currents (see Fig. 2.2 and related
    • Vortex-shedding regimes 85 M 1&> GN 5^ B N ON .M ±M T i ~ M # ^ p N 8^ time timeFigure 3.11 KC = 11. Lift-force time series obtained simultaneously in the same experiment as the flow visualization study of vortex motions depicted in Fig. 3.6a, which are reproduced here for convenience. The vertical arrows refer to cylinder motion. In the lift-force time series, the peaks marked A and C are caused by the growth and shedding of Vortex N and M (Frames 1 and 4) respectively, while the peak marked B is caused by the return of Vortex N towards the cylinder just after the flow reversal (Frame 3). Williamson (1985).
    • 86 Chapter S: Flow around a cylinder in oscillatory flowsdiscussion, Section 2.3). T h e positive peaks, on the other hand, (for example t h a tmarked B) are induced by the r e t u r n of the most recently shed vortex towards thecylinder just after flow reversal such as N in Frame 3. (The fact t h a t the cylinderexperiences a positive lift force when there is a vortex moving over the cylinderin the fashion as in Frame 3 was shown also by the theoretical work of Maulland Milliner (1978)). As is seen, not all the peaks in the lift force time series areinduced by the vortex shedding. v B N t 3 t 5 9 M6" "3& F v* % p ;^ Q^ 3-?H c Time Time Figure 3.12 15 < KC < 24. Lift-force time series obtained simultaneously in the same experiment as the flow visualization study of vor- tex motions depicted in Fig. 3.8, which are reproduced here for convenience. Williamson (1985).
    • Vortex-shedding regimes 87 W h e n closely examined, Fig. 3.12 will also indicate t h a t the peaks markedA, B, D, E, G and H are caused by vortex shedding, while those marked C and Fare induced by the r e t u r n of the most recently shed vortex towards the cylinderjust after the flow reversal, as described in the previous example. N, = - ^ Figure 3.13 Power spectra of lift. The quantity a is the variance of the lift fluctuations. Re = 5 X 10 5 . Justesen (1989). As a rule, we may say that the peak in the lift force which occurs just afterthe flow reversal is related to the return of the most recently shed vortex to thecylinder, while the rest of the peaks in the lift variation is associated with thevortex shedding. So, it is evident t h a t , in oscillatory flows, t h e lift-force frequencyis not identical to the vortex-shedding frequency. One way of determining the lift frequency is to obtain the power spectrumof the lift force and identify the dominant frequency. This frequency is called thef u n d a m e n t a l lift f r e q u e n c y . Fig. 3.13 gives an example; a sequence of powerspectra obtained for different values of KC number in an experiment where Re ismaintained constant at Re = 5 x 10 5 are given. Here, d>i, and a2 are the powerspectrum and the variance of the lift force, respectively. As seen, the fundamentallift frequency normalized by the oscillatory-flow frequency, namely
    • 88 Chapter S: Flow around a cylinder in oscillatory flows NL Ik (3.13) JWis 2 (that is, two oscillations in the lift force per flow cycle) for KC = 7; 9; 11; and12.5, while it switches t o 3 at t h e value of KC somewhere between KC = 12.5 and14 and is maintained at 3 (that is, three oscillations in the lift force per flow cycle)for KC = 14 and 16. T h e actual time series of the lift force corresponding to thespectrum for KC = 16 in t h e previous figure is given in Fig. 3.14, to illustratefurther the relation between the actual lift-force time series and the correspondingspectrum. u 1 1 / / imt 0 ft / -1 V / 2n /6rc 4FL(N) 100 ; cot -100 - 1st 2nd 3rd oscill. oscill. oscill. Three oscillations in FL per flow cycle (NL=3) Figure 3.14 Time series for the lift force corresponding to the spectrum for KC = 16 in Fig. 2.13. Re = 5 x 10 5 . Justesen (1989).
    • Effect of Reynolds number on flow regimes 89 Williamsons work (1985), where the ratio of Re to KC was maintainedconstant at j3 = Re/KC = 255 in one series of the tests and at 730 in the other,has indicated t h a t the fundamental lift frequency increases with increasing KC,as shown in Table 3.1. In these experiments, t h e KC number at which Ni switches from 2 to 3 isKC = 15, in contrast to the observation m a d e in Fig. 3.13 where Ni switchesfrom 2 to 3 at KC of about 13. This slight difference with regard to the KCnumber is related to the Reynolds number dependence. Table 3.1 Fundamental lift frequencies observed in the experiments of Williamson (1985). Normalized fundamental KC regime KC range Reynolds lift frequency number (= the number of Re oscillations in the lift per flow cycle) Single pair 7 < KC < 15 1.8-3.8 x 103 2 Double pair 15 < KC < 24 3 . 8 - 6 . 1 x 103 3 3 Three pairs 24 < KC < 32 6 . 1 - 8 . 2 x 10 4 Four pairs 32 < KC < 40 8 . 2 - 1 0 x 103 53.3 Effect of Reynolds number on flow regimes T h e detailed picture of the flow regimes as functions of b o t h the KC numberand the Re number is given in Figs. 3.15 and 3.16. Fig. 3.15 describes the role ofRe for small KC numbers (KC < 3). T h e figure illustrates how the boundariesbetween the different flow regimes, as described in Fig. 3.2, vary as a function ofRe. Furthermore, t h e following points may be m a d e with regard to Fig. 3.15:
    • 90 Chapter S: Flow around a cylinder in oscillatory flows turbulence RexlO" Figure 3.15 Regimes of flow around a smooth, circular cylinder in oscillatory flow for small KC numbers (KC < 3). (For large KC numbers, see Fig. 3.16). Explanation of various flow regimes in this figure: a: No separation. Creeping flow, a: No separation. Boundary layer is turbulent, b: Separation with Honji vortices (Fig. 3.3). c: A pair of symmetric vortices, d: A pair of symmetric vortices, but turbulence over the cylinder surface. Data: Circles from Sarpkaya (1986a); crosses for Re < 1000 from Honji (1981) and crosses for Re > 1000 from Sarpkaya (1986a). The diagram is adapted from Sarpkaya (1986a). 1) T h e curves which represent the inception of separation in Fig. 3.15 mustbe expected to approach asymptotically to the line Re = 5, as KC — oo (steady >current), to reconcile with the steady current case depicted in Fig. 1.1. 2) For large Re numbers (larger t h a n about 4 x 10 3 ), the non-separated flowregime may re-appear with an increase in the KC number, after t h e Honji type
    • Effect of Reynolds number on flow regimes 91separation has taken place (Fig. 3.15, Region a ) . This is linked with t h e transitionto turbulence in the b o u n d a r y layer. Once the flow in t h e b o u n d a r y layer becomesturbulent, this will delay separation and therefore the non-separated flow regimewill be re-established. However, in this case, t h e non-separated flow will be nolonger a purely viscous, creeping type of flow, b u t rather a non-separated flowwith turbulence over t h e cylinder surface. T h e transition to separated flow, onthe other hand, occurs directly with t h e formation of a pair of symmetric vortices(Region d, in Fig. 3.15). 200- i i i i i i 111 i i i i i i i M 100- Sarpkaya (1976a) KC 20 10- A pair of asymmetric vortices ~r See Pig 3.55 t -V ~—I I II 11II— 1 I llll| I I I I I I I ? 10° 10 4 10° Re 10° Figure 3.16 Vortex-shedding regimes around a smooth circular cylinder in oscillatory flow. Data: Lines, Sarpkaya (1976a) and Williamson (1985) and; squares from Justesen (1989). The quantity NL is the number of oscillations in the lift force per flow cycle: Ni = fhlfw in which fi is the fundamental lift frequency and fw is the frequency of oscillatory flow. Regarding t h e effect of Re for larger KC numbers (KC > 3) depicted inFig. 3.16, t h e presently available d a t a are not very extensive. It is evident thatno detailed account of various upper Reynolds-number regimes, known from thesteady-current research (such as the lower transition, the supercritical, the uppertransition a n d t h e transcritical regimes), is existent. Nevertheless, Sarpkayas
    • 92 Chapter 3: Flow around a cylinder in oscillatory flows(1976a) extensive d a t a covering a wide range of KC for lower Re regimes alongwith Williamsons (1985) and Justesens (1989) d a t a may indicate what happenswith increasing the Reynolds number. Regarding the vortex-shedding regimes, it is evident from t h e figure thatthe curves begin to bend down, as Re approaches to t h e value 10 5 , meaning thatin this region t h e normalized lift frequency Ni increases with increasing Re. Thisis consistent with the corresponding result in steady currents, namely t h a t theshedding frequency increases with increasing Re at 3.5 x 10 5 when the flow isswitched from subcritical to supercritical through the critical (lower transition)flow regime (Fig. 1.9). Finally, it may be mentioned t h a t Tatsumo and B e a r m a n (1990) presentedthe results of a detailed flow visualization study of flow at low KC numbers andlow /?(= Re/KC) numbers.3.4 Effect of wall proximity on flow regimes T h e influence of wall proximity on the flow around and forces on a cylinderhas already been discussed in t h e context of steady currents (Sections 1.2.1 and2.7). As has been seen, several changes occur in the flow around t h e cylinder whenthe cylinder is placed near a wall, such as the break-up of symmetry in t h e flow,the suppression of vortex shedding, etc.. T h e purpose of the present section is to examine the effect of wall proxim-ity on the regimes of flow around a cylinder exposed to an oscillatory flow. T h eanalysis is mainly based on the work of Sumer, Jensen and Freds0e (1991) wherea flow visualization study of vortex motions around a smooth cylinder was carriedout along with force measurements. T h e .Re-range of the flow-visualization experi-ments was 10 3 —10 4 , while t h a t of the force measurements was 0.4 x 10 5 —1.1 x 10 5 .Flow regimes4 < KC < 7 Fig. 3.17 illustrates how the vortices evolve during t h e course of the oscil-latory motion for KC = 4 for three different values of the gap-to-diameter ratioe/D, e being the gap between the cylinder and the wall. T h e symmetry observedin the formation and also in the motion of the vortices (Fig. 3.17a) is no longerpresent when e/D = 0.1 (Fig. 3.17b). This is also clear from t h e lift-force tracesgiven in Fig. 3.18 where almost no lift force is exerted on the cylinder for e/D = 2,while a non-zero lift exists for e/D = 0 . 1 . Here Ci is the lift coefficient defined by
    • Effect of wall proximity on flow regimes 93 a) — ^ cot =90° - ~ 135° — 158° - ^ 180° eg b) 135" GL3 77777777777777777 158° 180" K & 2) 77777777777777777 77777777777777777 c) cot = 0 60" K >5> 77777^7777/ 7777777)l777777777 120" 180° 77777777777777777 7777777^,77777777Figure 3.17 Evolution of vortex motions. KC = 4. Gap-to-diameter-ratio values: (a) e/D = 2, (b) e/D = 0.1, (c) e / D = 0. Sumer et al. (1991).
    • 94 Chapter S: Flow around a cylinder in oscillatory flows Fy = -pCLDUl (3.14) T h e vortex regime is quite simple for the wall-mounted cylinder (Fig. 3.17c):a vortex grows behind the cylinder each half-period, and is washed over the cylinderas the next half-period progresses. Jacobsen, B r y n d u m and Freds0e (1984) givea detailed account of the latter where the motion of the lee-wake vortex over thecylinder is linked to the maximum pressure gradient in the outer flow. T h e lift-force trace is presented in Fig. 3.18c. T h e peaks in the lift force are associatedwith the occurrences where the vortices (Vortex K, Vortex £,... in Fig. 3.17c) arewashed over the cylinder. Velocity, U(t) 0 360 Figure 3.18 Lift-force traces. KC = 4. Sumer et al. (1991).
    • Effect of wall proximity on flow regimes 95
    • 96 Chapter 3: Flow around a cylinder in oscillatory flows oot=0° 75° M ////////WW /s/ssssss/s/s; ////////////// 90° 135° 150° ;//;/////////; s///////////// ////////////// <ot=0° 40° 75° CO /S///7/7777777 a /7/77777777//7 W//W///777 90° 120° 150° M T OPQ /////77777T777 77/77777777777 a>t = 10 40 75 KG, Q ^ 77777777777777 Jy^ •77777777777777 /MW//////// 90 120 150 M - ^ M 1V1 ? L >r 7^777777777777 ?s7/S/////, 77777777777777 Figure 3.20 Evolution of vortex motions in the range 7 < i f C < 15. In the tests presented here KC = 10. Gap-to-diameter-ratio values: (a) e/D = 1, (b) e/D = 0.1, (c) e/D = 0. Sumer et al. (1991).
    • Effect of wall proximity on flow regimes 977 < K C < 15 One of the interesting features of this KC regime for a wall-free cylinderis the formation of t h e transverse vortex street where the shed vortices form avortex street perpendicular to the flow direction (Figs. 3.6a and 3.7). Sumer etal.s (1991) work shows t h a t the transverse street regime disappears when the gapbetween the cylinder and the wall becomes less than tibout 1.7-1.8 times the cylinderdiameter. Figs. 3.19a and 3.19b illustrate two different vortex flow regimes, onewith a gap ratio above this critical value (the transverse street regime) and theother below it, where the transverse vortex street is replaced by a wake regionwhich lies parallel to t h e flow oscillation direction. (a) e / D = 1. Fig. 3.20a illustrates the time development of vortex motionsduring one half-period of the motion, while Fig. 3.21b presents the correspondinglift-force trace. Fig. 3.20a indicates t h a t there is only one vortex shed (Vortex L)during one half-period of the motion. Fig. 3.21b shows how the lift force evolvesduring the course of the motion. T h e negative peak (B in Fig. 3.21b) is caused bythe development of Vortex K (Fig. 3.20a, cot = 0° - 45°) (see Maull a n d Milliner(1978) for the relation between the vortex motion and the forces). As VortexK is washed over t h e cylinder, the cylinder experiences a positive lift force, andthe development of Vortex L also exerts a positive lift force (C in Fig. 3.21b).As Vortex L moves away from t h e cylinder (wi = 135° — 150°), t h e positive liftexerted on the cylinder by Vortex L is diminished. (b) e / D = 0 . 1 . T h e main difference between this case and the previous oneis that here the wall-side vortex (Vortex TV) grows quite substantially. It is thislatter vortex which is washed over the cylinder, whereas in the former case it wasthe free-stream-side vortex (Vortex M). T h e positive peak in the lift force (D in Fig. 3.21c) is caused by the de-velopment of Vortex L. T h e negative peak in t h e lift force (E in Fig. 3.21c) iscaused by the development of Vortex T combined with t h e high velocities in the Vgap induced by t h e flow reversal. (c) e / D = 0. In this case, the vortex which develops behind the cylinderin the previous half-period (Vortex K in Fig. 3.20c) a n d t h e vortex which isnewly created (Vortex L in Fig. 3.20c) form a vortex pair. This pair is then setinto motion owing to its self-induced velocity field, and t h u s steadily moves awayfrom the cylinder in t h e downstream direction (see Fig. 3.20c, urt = 40° — 120°).Following t h e removal of Vortex L, a new vortex (Vortex M) begins t o developbehind the cylinder. T h e visualization results show t h a t the way in which the vortex flow regimedevelops for the wall-mounted cylinder ( e / D = 0) remains the same, irrespectiveof the range of KC. It should be noted, however, t h a t the individual events suchas the formation of t h e vortex pair etc. may occur at different phase (tvt) valuesfor different KC ranges. T h e peaks in t h e lift-force trace are caused by the passage of Vortex K overthe cylinder.
    • 98 Chapter S: Flow around a cylinder in oscillatory flows Velocity, U(t) 7I7Z /////*////// 7 < KC < 13 a) % 13 < KC < 15 b) £ - l c) H =0.1 d) 0 Figure 3.21 Lift-force traces in the range 7 < KC < 15. Positive lift is directed away from the wall. The wall-free cylinder traces (a), e/D = oo, are taken from Williamson (1985). For the tests presented here KC = 10. Sumer et al. (1991).
    • Effect of wall •proximity on flow regimes 9915 < K C < 2 4 a n d f u r t h e r K C r e g i m e s First, the KC regime 15 < KC < 24 will be considered. (a) e / D = 1. In this KC regime for wall-free cylinders there is no symmetrybetween the half-periods, as far as the vortex motions are concerned (Figs. 3.8and 3.12), and this also applies to t h e present case where e/D = 1, as seen fromFig. 3.22a; t h e vortex which is washed over the cylinder alternates between thewall side and the free-stream side each half-period. T h e lift-force variation (Fig.3.23b) supports this asymmetric flow picture. (b) e / D = 0 . 1 . Here, the flow is asymmetry; it is always the wall-sidevortex (Vortex P , Fig. 3.22b) which is washed over the cylinder before the flowreverses to start a new half-period. T h e lift force is directed away from t h e wall most of the time (Fig. 3.23c).Furthermore, it contains distinct, short-duration peaks in its variation with time(F, G in Fig. 3.23c). T h e flow-visualization tests show t h a t these peaks areassociated with t h e vortex shedding at the wall side of the cylinder: such peaksoccur whenever there is a growing vortex on t h a t side of the cylinder (Fig. 3.22b:wi = 50° - 60° and ut = 80° - 93°). Fig. 3.24 represents t h e lift-force traces separately for t h e interval 0.05 <e/D < 0.4. For values of the gap ratio smaller t h a n approximately 0.3, the liftforce becomes asymmetric, being directed away from the wall for most of t h etime, containing t h e previously mentioned distinct short-duration peaks. Thesepeaks are present even for the gap ratio e/D = 0.05. These short-duration peaksindicate t h a t t h e vortex shedding is maintained even for very small gap ratiossuch as e/D = 0.1, in contrast to what occurs in steady currents where the vortexshedding is maintained for values of gap ratio down to only about e/D = 0.3(Section 1.2.1, Fig. 1.21). This aspect of the problem will be discussed in greaterdetail later in this section. (c) e / D = 0. It is apparent from Fig. 3.22c t h a t t h e m a n n e r in which t h evortex flow regime develops is exactly the same as in t h e range 7 < KC < 15 (cf.Figs. 3.20c and 3.22c). However, the streamwise distance t h a t the vortex pairtravels is now relatively larger. T h e lift force (Fig. 3.23d) varies with respect to time in the same way asin Fig. 3.21d where 7 < KC < 15. However, the peaks in t h e present case occurrelatively earlier t h a n those in Fig. 3.21d. T h e visualization tests of Sumer et al. (1991) indicate t h a t , as in Williamson(1985), t h e flow p a t t e r n s for the KC regimes beyond KC = 24 differ only in t h enumber of vortices shed with no basic changes in the actual flow p a t t e r n s .Vortex shedding W h e t h e r vortex shedding will be suppressed for small values of the gapratio can be detected from t h e flow-visualization films as well as from t h e lift-forcetraces. T h e results of such an analysis are plotted in Fig. 3.25. From t h e figure,the following observations can be made.
    • 100 Chapter 3: Flow around a cylinder in oscillatory flows O o cot = -10 . 60 120 K (jj/ K M V; 7CD " ^ 7777777777777777- Jf/77/J?J?/7???} w/«w/«wr o 0 205° 275 350 M Gj & 3 i*cp 9 o cot= 10° 50° 60 — K 7777777777777777 7/JJM77M77/77 o o o 80 93 140 •7777777777777777 7777777777777777 7777/77777777777 cot= 10 40 75 M 7777777777777777 L 7777777777777777 go 7777777777777 77777777777 L-l 90 150 M M K 77777777777777777777777777777? 7777777777777777 Figure 3.22 Evolution of vortex motions in the range 15 < AC < 24. In the tests presented here KC = 20. Gap-to-diameter-ratio values: (a) e/D = 1, (b) e/D = 0.1, (c) e/D = 0. Sumer et al. (1991).
    • Effect of wall proximity on flow regimes 101 Velocity, U(t) 0/ 360/ tot V77777 0 n =01Figure 3.23 Lift-force traces in the range 15 < KC < 24. Positive lift is directed away from the wall. The wall-free cylinder (e/D = oo) trace (a) is taken from Williamson (1985), see Fig. 3.12. In the tests presented here KC = 20. Sumer et al. (1991).
    • 102 Chapter S: Flow around a cylinder in oscillatory flows Velocity, U(t) 0 / ~ 360". /////X/S//// 7Z77. a) § = 0.4 b) D 0 % = 0.1 d) % = 0.05 Figure 3.24 Lift-force traces for the ranges 0.05 < e/D < 0.4 and 15 < KC < 24. Positive lift is directed away from the wall. In the tests presented here KC = 20. (a) e/D = 0.4, (b) e/D = 0.2, (c) e/D = 0.1, (d) e/D = 0.05. Sumer et al. (1991).
    • Effect of wall proximity onflow regimes 103 1) For large values of KC, it appears t h a t the gap ratio below which thevortex shedding is suppressed approaches the critical value e/D « 0.25 deducedfrom the work by B e a r m a n and Zdravkovich (1978) and by Grass et al. (1984) forsteady currents, (Section 1.2.1). 2) Although t h e borderline between the two regions in the figure, namelyt h e vortex-shedding region and the region where the vortex shedding is suppressed,is not expected to be a clean-cut curve, there is a clear tendency t h a t the vortexshedding is maintained for smaller and smaller values of the gap ratio as KC isdecreased. Vortex shedding is maintained even for very small gap ratios such as e/D =0.1 for KC = 10 — 20, as shown in the photograph in Fig. 3.19c. Likewise, Fig.3.24c implies t h a t shedding occurs for t h a t value of t h e gap ratio, as t h e short-duration peaks in the lift-force time series are associated with vortex shedding.T h e reason why vortex shedding is maintained for such small gap ratios is becausethe water discharge at the wall side of the cylinder is much larger in oscillatoryflow at small KC t h a n in steady currents due to the large pressure gradient fromthe wave. D Vortex 0.4 • • • shedding 0.2 l Vortex I OA O A O A shedding ,—— o O O O OA O O A suppressed O O A O —I _L_ _L_ 20 40 oo (steady KC current) Figure 3.25 Diagram showing where the vortex shedding is suppressed in the (e/D, iirC)-plane. Open symbols: vortex shedding is sup- pressed. Filled symbols: vortex shedding exists, o, A , experi- ments of Sumer et al. (1991). (o from flow visualization, A from lift-force traces); a, Bearman and Zdravkovich (1978); /, Grass et al. (1984).
    • 104 Chapter S: Flow around a cylinder in oscillatory flows T h e frequency of vortex shedding can be defined by an average frequencybased on the number of the short-duration peaks in the lift force over a certainperiod, as sketched in Fig. 3.26. T h e figure depicts t h e Strouhal number, basedon this frequency and the maximum flow velocity S* = JT (3-15)as a function of the gap ratio. T h e shedding frequency actually varies over thecycle. T h e /„-value used in the definition of St in the preceding equation isaveraged over a sufficiently long period of time. Fig. 3.27 presents the same d a t ain the normalized form St/Stg where Sto is the value of St attained for large valuesof e/D. Also plotted in Fig. 3.27 are the results of two studies conducted in steadycurrents, namely Grass et al. (1984) and Raven et al. (1985). T h e details regardingthese two latter studies have already been mentioned in the previous chapter (seeFig. 1.23 and t h e related text). From Figs. 3.26 and 3.27 t h e following conclusionscan be drawn. 1) For a given e/D, St increases (albeit slightly) with decreasing KC (Fig.3.26). 2) T h e measurements collapse remarkably well on a single curve when plot-ted in the normalized form S i / S i o v e r s u s e/D (Fig. 3.27), where the influence ofthe close proximity of the wall on St can be seen even more clearly. 3) It is apparent t h a t St increases as t h e gap ratio decreases. T h e increasein St frequency can be considerable (by as much as 50%) when the cylinder isplaced very near the wall (e/D = 0.1 — 0.2). This is because t h e presence of thewall causes the wall-side vortex to be formed closer to t h e free-stream-side vortex.As a result of this, the two vortices interact at a faster rate, leading to a higherSt frequency. Finally, Sumer et al.s (1991) work indicates t h a t there is almost no notice-able difference between the shedding frequency obtained in their smooth-cylinderexperiments and t h a t obtained in their supplementary experiments with a roughcylinder (the cylinder roughness in the latter experiments is about k3/D = 10~ 2 ).3.5 Correlation length It has been seen that vortex shedding around a cylinder occurs in cellsalong the length of the cylinder (Section 1.2.2), and t h a t the spanwise correlationcoefficient is one quantity which gives information about t h e length of these cells.T h e studies concerning the effect of Re number, the effect of cylinder vibration,and the effect of turbulence in the incoming flow on correlation in steady currentshave been reviewed in Section 1.2.2. In the present section, we will focus on thecorrelation measurements m a d e for cylinders exposed to oscillatory flows.
    • Correlation 105 0.4 bo •Da o A 2 0.2 9 9 4 Vortex i shedding 0 J L 0 1 2 e/D Figure 3.26 Strouhal number versus gap ratio, o, KC = 20; A , KC = 30; a, KC = 55; V . AC = 65. Sumer et al. (1991). These measurements have been m a d e by Obasaju, Bearman and G r a h a m(1988), Kozakiewicz, Sumer and Freds0e (1992) and Sumer, Freds0e and Jensen(1994). Obasaju et al.s (1988) study has clearly demonstrated t h a t the correlationis strongly dependent on the Keulegan-Carpenter number. Fig. 3.28 depicts theirresults, 2 being the spanwise separation (see Eq. 1.10). In t h e study of Obasaju etal., the correlation measurements were m a d e by measuring the pressure differential,i.e. the difference between the pressures on the diametrically opposite points at thetop and b o t t o m of the cylinder. Fig. 3.28 indicates t h a t the correlation coefficienttakes very large values when KC is small, while it takes t h e lowest value whenKC is at about 22. Obasaju et al. (1988) give a detailed accoint of the behaviourof the correlation coefficient as a function of the KC number. They link the lowcorrelation measured at KC = 22 to the fact t h a t KC = 22 lies at the boundarybetween t h e two AC-regimes, 15 < KC < 24 and 24 < KC < 30, while theyargue t h a t the correlation is measured to be high at KC = 10 because KC = 10lies in the center of the AC-regime 7 < KC < 15 (see also Bearman, 1985). Fig. 3.29 illustrates t h e time evolution of the correlation coefficient for agiven value of the spanwise separation distance, namely z/D = 1.8, as the flow
    • 106 Chapter S: Flow around a cylinder in oscillatory flows St St, Vortex shedding e/D Figure 3.27 Normalized Strouhal number as function of gap ratio, o, KC = 20; A , KC = 30; •, KC = 55; V , KC = 65; x , steady current (Raven et al., 1985); - -, steady current (Grass et al., 1984). Sumer et al. (1991).progresses. Here KC = 65, and the figure is taken from Kozakiewicz et al.s (1992)study where the cylinder was placed at a distance from a plane wall with the gap-ratio e/D = 1.5, sufficiently away from t h e wall so t h a t the wall effects could beconsidered insignificant. T h e correlation coefficient is calculated from the signals received from thepressure transducers mounted along the length of the cylinder using t h e followingequations, Eqs. 3.16 and 3.20): P(C, o r t M C + z, wQ R(z, ujt) (3.16) b (C, ^)] 1 / 2 b 2 (C + ^ ^ ) ] 1 / 2 2in which ( is t h e spanwise distance, z is t h e spanwise separation between twopressure transducers, and p is t h e fluctuation in pressure defined by p=p-p (3.17)the pressure transducers being at the free-stream-side of the cylinder. T h e overbar in t h e preceding equations denotes ensemble averaging:
    • Correlation 107 R I ^ K C = 10 1.0 N^ - * ^ C 2 T ~ ° • "--«--• 0.8 0 * . 18 **"" 0.6 0.4 26 • ~~~~-t**~^~ • -~-> + -34 0.2 a - ~ ——JVZ^ " 4 2 i l 1 — 1 — e ^ H S -22 8 z/D Figure 3.28 Average values of correlation coefficients versus spanwise sepa- ration. (a) V , KC = 10; o, 18; *, 18; D, 22; A , 26; + , 34; ., 42. Note /?(= Re/KC) = 683 except for the case denoted by * where fi = 1597. Obasaju et al. (1988). M 1 w (3.18) P=I?£PK> (* + 0-i):r)] J=l M p2 = i E M c ^ ( + (j-i)T)]} 2 (3.19) i=i M p(C, U*)P(C + *, ci) = — J^plC, u{t+(j-l)Tp[C+z, u,(t+(;-l)T] (3.20)in which T is the period of the oscillatory flow, and M is t h e total number of flowcycles sampled. Fig. 3.29 shows t h a t the correlation coefficient increases towards the endof every half period, and attains its m a x i m u m at the phase tot = 165°, about 15°before the outer flow reverses. This phase value corresponds to the instant wherethe flow at the measurement points comes to a standstill, as can be traced fromthe pressure traces given in Kozakiewicz et al. (1992). As the flow progresses fromthis point onwards, however, the correlation gradually decreases and assumes its
    • 108 Chapter S: Flow around a cylinder in oscillatory flows R(»t)z = !. 8 D Figure 3.29 Correlation coefficient as a function of phase ujt. KC = 65, Re — 6.8 X 10 4 , e/D = 1.5 (sufficiently large for the wall effects to be considered insignificant), z/D = 1.8. Kozakiewicz et al. (1992).minimum value for some period of time. T h e n it increases again towards the endof t h e next half period. Fig. 3.30 shows three video sequences at the phase values u>t = 113°, 165°and 180°. T h e flow picture in Fig. 3.30b shows t h a t the shear layer markedby the hydrogen bubble has rolled up into its vortex (A in Fig. 3.30b) and isstanding motionless. As time progresses from this point onwards, however, thisvortex begins to move in the reverse direction and is washed over the cylinder as acoherent entity along t h e length of the cylinder (Fig. 3.30c). Now, comparison ofFig. 3.30a with Fig. 3.30b indicates t h a t while spanwise cell structures can easilybe identified in the former (ut = 113°), no such structure is apparent in Fig. 3.30b(u>t = 165°), meaning t h a t t h e spanwise correlation should be distinctly larger inthe latter t h a n in the former case. T h e same is also t r u e for cot = 180° where,again, large correlations should be expected. This is indeed the case found in thepreceding in relation to Fig. 3.29. Effect of w a l l p r o x i m i t y o n c o r r e l a t i o n Kozakiewicz et al.s (1992) study covers also the near-wall cylinder case. Fig. 3.31 shows t h e correlation coefficients for four different test d a t a with e/D = 2.3, 1.5, 0.1 and 0 where e is the gap between the wall and t h e cylinder.
    • Figure 3.30 Hydrogen-bubble flow visualization sequence of pictures showing the time d cell structures for a stationary cylinder. D = 2 cm, KC = 40, Re = 2 (1992). The cylinder is located well away from a wall, namely the gap-t therefore, the effect of wall proximity could be considered insignificant.
    • 110 Chapter 8: Flow around a cylinder in oscillatory flows 1 (c) KC = 65 R aD w e WW/////// + 0.2 sv x+ "*"*" g 1 -*-*- f £l 1 8 z/D Figure 3.31 Period-averaged correlation coefficient. Wall proximity effect re- garding the pressure fluctuations. See Fig. 3.32 for the wall proximity effect regarding the correlation of the lift force. Koza- kiewicz et al. (1992).
    • Correlation 111 T h e correlation coefficients presented in Fig. 3.31 are the period-averagedcorrelation coefficient, which is defined by 1 /27r R(z) = — / R(z, ut) d{ojt) (3.21) 2TT J0 T h e general trend in Fig. 3.31 is t h a t the correlation coefficient decreaseswith decreasing gap ratio. However, caution must be exercised in interpretingthe results in the figure. While R for e/D = 2.3 and 1.5 can be regarded as thecorrelation coefficient also for t h e lift force on the cylinder (since the fluctuationsp for which R is calculated are caused by the vortex shedding), this is not thecase for e/D = 0.1 and 0. First of all, for e/D = 0, t h e vortex shedding istotally absent (Fig. 1.21), and t h e fluctuations in the measured pressure, p, inthis case degenerate from those induced by the highly organized vortex-sheddingphenomenon (e/D = 2.3 and 1.5) to those due to disorganized turbulence. So,the correlation, R, for this case, namely e/D = 0, only give information about thelength scale in the spanwise direction of this turbulence. For e/D = 0.1, on t h e other hand, the vortex shedding may be maintainedparticularly for small KC numbers (see Fig. 3.25). However, the lift in thiscase consists of two p a r t s , a low frequency portion which is caused by the closeproximity of the wall and the superimposed high-frequency fluctuations which arecaused by vortex shedding (Fig. 3.23c). As such, t h e correlation, R, calculated onthe basis of fluctuations, p, which are associated with the vortex shedding only,cannot be regarded as the correlation coefficient also for the lift force for the caseof e/D = 0.1. Regarding the correlation of the lift force itself, Kozakiewicz et al. (1992)did some indicative experiments for t h e wall-mounted cylinder situation with thepressure transducers positioned on the flow side of the cylinder. Clearly, with thisarrangement t h e pressure time-series can be substituted in place of t h e lift forceones, as far as t h e correlation calculations are concerned. Regarding the lift forceitself, t h e lift in this case (e/D = 0) is not caused by the pressure fluctuations (asopposed to what occurs in the case of a wall-free cylinder, Fig. 3.23a), but ratherby t h e contraction of t h e streamlines near t h e flow side of t h e cylinder as well asby the movement of t h e lee-wake vortex over the cylinder, which results in theobserved peak in the lift force prior to the flow reversal in each half-cycle of themotion (Fig. 3.23d). Hence, the correlation in connection with t h e lift force in thiscase cannot be calculated by Eq. 3.16 (which is based on the pressure fluctuationsrather t h a n on t h e pressure itself); instead, t h e usual time-averaging should beemployed, i.e. the correlation is calculated by Eq. 1.10. Fig. 3.32 presents the spanwise correlation coefficients obtained for thewall-mounted cylinder, where the results for e/D = 2.3 of Fig. 3.31 are replottedto facilitate comparison. T h e correlations in these diagrams are now all associatedwith the lift force; therefore comparison can be m a d e on t h e same basis. T h e figureindicates t h a t , as expected, t h e correlation increases tremendously as the gap ratiochanges from 2.3 (the wall-free cylinder) to nil (the wall-mounted cylinder).
    • 112 Chapter 3: Flow around a cylinder in oscillatory flows R,RT A K C = 6 /RT ^D: D r-o-o-o-<v&-o-o-o- 0 *n**~* R 2.3 ////A//// 0.2 0 J L 8 z/D 8 z/D z/D Figure 3.32 Correlation coefficient for the lift force on cylinder, showing wall proximity effect. Rx for the wall-mounted cylinder is computed direct from pressure signals employing time-averaging according to Eq. 1.10. Kozakiewicz et al. (1992).
    • Correlation 113Effect of v i b r a t i o n s o n c o r r e l a t i o n This section focuses on the effect of vibrations on the correlation when thecylinder is vibrated in a direction perpendicular to the flow only. Fig. 3.33 presentsthe correlation coefficients as functions of the double-amplitude-to-diameter ratiofor three KC numbers, Kozakiewicz et al. (1992). In the study of Kozakiewicz etal., the vibrations were not free, but rather forced vibrations. Also, the cylindervibrations were synchronized with t h e outer oscillatory-flow motion. T h e resultsof Fig. 3.33 may be compared with t h e corresponding results of Novak and Tanaka(1977) obtained for steady currents (Fig. 1.28). Note t h a t in Novak and Tanakasstudy t h e cylinder is vibrated with a frequency equal to its vortex-shedding fre-quency, which is identical to the fundamental lift frequency. Likewise, in t h e studypresented in Fig. 3.33, the cylinder is vibrated with a frequency equal to the fun-damental lift frequency. If this frequency is denoted by / L and t h e frequency ofthe oscillatory flow by / „ , then NL = fhlfw w m become the number of oscilla-tions in the lift force for one cycle of the flow as discussed in Section 3.2 (see Eq.3.13). In Kozakiewicz et al.s study NL was set equal to 13 for KC = 65, to 4for KC = 20, and to 2 for KC = 6. Note t h a t these figures are in accordancewith Sarpkayas (1976a) stationary-cylinder lift-force frequency results (Fig. 3.16)and also with Sumer and Freds0es (1988) results with regard to the cross-flowvibration frequency of a flexibly-mounted cylinder subject to an oscillatory flow. Returning t o Fig. 3.33, the following conclusions can be deduced from thefigure: 1) A constant increase in the correlation coefficient with increasing ampli-tudes takes place u p to t h e values of 2A/D of about 0.2 for KC = 6 a n d u p to2A/D = 0.3 for KC = 20 and 65. This can be seen even more clearly from Fig.3.34 where the correlation coefficient at the spanwise distance z = D is plotted asa function of 2A/D. T h e way in which the correlation coefficient increases withincreasing amplitude-to-diameter ratio is in accord with the steady current results(Fig. 3.34d). However, this increase is not as large as in steady currents. 2) T h e correlation decreases, however, for further increase in t h e value of2A/D. This may be a t t r i b u t e d to t h e change in the flow regime with increasing2A/D (this change in the flow regime with increasing 2A/D has been demonstratedby Williamson and Roshko (1988) for a cylinder exposed to a steady current). Nopressure correlation d a t a are available for the steady-current situation for valuesof 2A/D larger t h a n 0.25. Therefore, no comparison could be m a d e as far assuch high values of 2A/D are concerned. There are, however, correlation mea-surements (Ramberg and Griffin, 1976) for 2A/D values as large as 0.7, wheret h e correlation coefficient is based on wake velocity signals; these measurementsindicate t h a t the correlation coefficient increases in a monotonous m a n n e r withincreasing amplitudes. In a subsequent study, Sumer et al. (1994) measured the correlation ona freely-vibrating cylinder. Their results indicated t h a t the correlation increasesmonotonously with increasing amplitude of vibrations (Fig. 3.35). T h e observed
    • Ill) Chapter S: Flow around a cylinder in oscillatory flows 6 z/D z/D z/D Figure 3.33 Period-averaged correlation coefficient for vibrating cylinder for e/D = 1.5. (a) NL = 2 and Re = 3.4 x 10 4 for KC = 6; (b) NL = 4 and Re = 6.8 X 10 4 for KC = 20 and (c) NL = 13 and Re = 6.8 x 10 4 for KC = 65. Vibrations are forced vibrations and Ni being the normalized fundamental lift frequences (Eq. 3.13). Kozakiewicz et al. (1992).
    • Correlation 115 Cylinder with forced vibrations ..(a) — Steady current 0 0 0.2 0.4 0.6 2A/D Figure 3.34 (a), (b) and (c): Period-averaged correlation cofficient with re- spect to vibration amplitudes for different KC numbers; (d): Steady-current data (Howell and Novak, 1979), e/D = CO and Re = 7.5 x 10 4 . Vibrations are forced vibrations. Kozakiewicz et al. (1992).difference between t h e variation of correlation coefficients in the case of forcedvibrations (Fig. 3.34) and t h a t in the case of self-induced vibrations (the freely-vibrating-cylinder case, Fig. 3.35) is a t t r i b u t e d to the change in the phase betweenthe cylinder vibration and the flow velocity: In t h e tests of Kozakiewicz et al. (Fig. 3.34), t h e cylinder motion is synchro-nized with the outer, oscillatory-flow motion such t h a t the instants correspondingto t h e zero upcrossings in the outer-flow velocity time series coincide with the zerodowncrossings in the cylinder-vibration time series. In the tests of Sumer et al.(Fig. 3.35), however, the vibrations are self-induced, and apparently t h e phase be-tween the cylinder vibration and the flow velocity, rj>, is not constant, b u t rather afunction of the reduced velocity (Fig. 3.36). Obviously, any change in t h e quantity4> may influence t h e end result considerably. This may explain the disagreementbetween t h e results of Kozakiewicz et al.s study (Fig. 3.34) and those of Sumeret al.s study (Fig. 3.35). Fig. 3.37 illustrates how the vibration frequency influences the correlationcoefficient. Here Ni = 13 is the number of vibrations in one cycle of t h e oscillatory
    • 116 Chapter 3: Flow around a cylinder in oscillatory flows Freely-vibrating cylinder R z = 3D 0.8 o V r < 5.6 0.6 v v > 5.6 r KC = 10 0.4 0.2 0 J I I L 0 0.4 0.8 2A/D Figure 3.35 Period-averaged correlation coefficient with respect to vibra- tion amplitudes. Vr is the reduced velocity defined by Vr = Um/(Dfn) in which fn is the natural frequency of the flexibly- mounted cylinder. Vibrations are not forced, but rather self- induced vibrations. Sumer et al. (1994).flow, and it corresponds to the fundamental lift frequency corresponding to astationary cylinder. As is seen, R decreases as the vibration frequency movesaway from the fundamental lift-force frequency. This result is in agreement withthe corresponding result obtained in Toebes (1969) study for the steady-currentsituation.3.6 Streaming In the case of unseparated flow around the cylinder, a constant, secondaryflow in the form of recirculating cells emerge around the cylinder (Fig. 3.38).This is called streaming. A simple explanation for the emergence of this steadystreaming may be given as follows. T h e flow velocity experienced at any point near the surface of the cylinder(Point A, say, in Fig. 3.38) is asymmetric with respect to two consecutive halfperiods of the flow. Namely, the velocity is relatively larger when the flow is inthe direction of converging surface geometry t h a n t h a t when the flow is in theopposite direction, as sketched in Fig. 3.39 (this is due to the difference in theresponse of the cylinder boundary layer in the two half periods, namely in theconverging half period a n d in the diverging half period). This asymmetry in thevelocity results in a non-zero mean velocity in the direction towards t h e top in the
    • Streaming 111 J, u 1 l80° 360° <P° / y [ ^/ 100 T r f J z.0* 80 V 60 o 40 - o ° 0 20 o J I I I U 0 2 3 4 5 6 7 8 V,Figure 3.36 Phase difference between the cylinder vibration and the flow ve- locity in the tests presented in the previous figure. Sumer et al. (1994). l.Oi R NL: fc^TTr—9-13 *G3cJfr—*- 14 0.2 _ ^ " :j~~nHnTT 12 *" 6 0 1 I I I * . 0 8 z/DFigure 3.37 Effect of vibration frequency on period-averaged correlation coef- ficients for KC = 65, e/D = 1.5 and 2A/D = 0.25. Vibrations are forced vibrations. Kozakiewicz et al. (1992).
    • 118 Chapter 8: Flow around a cylinder in oscillatory flows Figure 3.38 Steady streaming around a cylinder which is subject to an oscil- latory, unseparated flow. Flow Flow from a t o b from b t o a Figure 3.39 Asymmetry in two consecutive half periods in the velocity at a point near the cylinder surface that results in a steady streaming towards the top of the cylinder.upper half of the cylinder and towards the b o t t o m in the lower half of the cylinder.This presumably leads to the recirculating flow p a t t e r n shown in Fig. 3.38.
    • Streaming 119 T h e streaming has been the subject of an extensive research with regard toits application in the field of acoustics (see Schlichting (1979, p.428) and Wang(1968)). It may be i m p o r t a n t also in the field of offshore engineering in conjunctionwith the sediment motion and t h e related deposition and scour processes aroundvery large, bottom-seated marine structures which are subject to waves. Figure 3.40 (a): The steady streaming caused by an oscillating circular cylin- der. Re = 2, KC = 3 X 1 ( T 2 . (b): The thickness of recirculat- ing cells, o, experiment, (Holtsmark et a]., 1954); —, theory by Wang (1968). Wang (1968) developed an analytical theory for very small Re numbers(creeping flow) and KC numbers. Wangs results compare very well with theexperiments. In t h e study, analytical expressions were obtained for t h e streamfunction and t h e drag coefficient. Fig. 3.40a shows the flow picture obtained by
    • 120 Chapter S: Flow around a cylinder in oscillatory flowsWang for Re = 2 and KC = 3 x 10~ 2 , while Fig. 3.40b depicts t h e variation ofthe thickness of the recirculating cells as a function of Re and KC numbers. For large Re numbers, apparently no study is available in t h e literature.Therefore it is difficult to make an assessment of the thickness of t h e recirculatingcells and the magnitude of the streaming. However, t h e results of Sumer, Laursenand Freds0es study (1993) on oscillatory flow in a convergent/divergent tunnel,where t h e Reynolds number was rather large (indeed, so large t h a t the boundarylayer was turbulent) suggest t h a t the thickness of the recirculating cell may be inthe order of magnitude of the boundary-layer thickness and the magnitude of thestreaming may be in the order of magnitude of O(0.1Um). In a recent study (Badr, Dennis, Kocabiyik and Nguyen, 1995), the solutionof N.-S. equations was achieved for Re = 10 3 and KC = 2 and 4. T h e time-averaged flow field over one period obtained by the authors revealed t h e presenceof the steady streaming p a t t e r n (depicted in Fig. 3.38) even in the case of separatedflow.REFERENCESBadr, H.M., Dennis, S.C.R., Kocabiyik, S. and Nguyen, P. (1995): Viscous oscil- latory flow about a circular cylinder at small to moderate Strouhal number. J. Fluid Mech., 303:215-232.Bearman, P.W. (1985): Vortex trajectories in oscillatory flow. Proc. Int. Symp. on Separated Flow Around Marine Structures. T h e Norwegian Inst, of Technology, Trondheim, Norway, J u n e 26-28, 1985, p. 133-153.Bearman, P.W. and G r a h a m , J.M.R. (1979): Hydrodynamic forces on cylindrical bodies in oscillatory flow. Proc. 2nd Int. Conf. on t h e Behaviour of Offshore Structures, London, 1:309-322.Bearman, P.W. and Zdravkovich, M.M. (1978): Flow around a circular cylinder near a plane boundary. J. Fluid Mech., 89:33-48.Bearman, P.W., G r a h a m , J.M.R., Naylor, P. and Obasaju, E.D. (1981): T h e role of vortices in oscillatory flow about bluff cylinders. Proc. Int. Symp. on Hydrodyn. in Ocean Engr., Trondheim, Norway, 1:621-643.Bearman, P.W., G r a h a m , J.M.R. and Singh, S. (1979): Forces on cylinders in harmonically oscillating flow. Proc. Symp. on Mechanics of Wave Induced Forces on Cylinders, Bristol, ed. T.L. Shaw, P i t m a n , p p . 437-449.
    • References 121Grass, A.J. and K e m p , P.H. (1979): Flow visualization studies of oscillatory flow past smooth and rough circular cylinders. Proc. Symp. on Mechanics of Wave-Induced Forces on Cylinders, Bristol, ed. T.L. Shaw, P i t m a n , pp. 406-420.Grass, A.J., Raven, P.W.J., Stuart, R.J. and Bray, J.A. (1984): T h e influence of b o u n d a r y layer velocity gradients and b e d proximity on vortex shedding from free spanning pipelines. Trans. ASME, J. Energy Resour. Tech., 106:70-78.Hall, P. (1984): On t h e stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech., 146:347-367.Holtsmark, J., Johnsen, I., Sikkeland, I. and Skavlem, S. (1954): Boundary layer flow near a cylindrical obstacle in an oscillating incompressible fluid. J. Acoust. Soc. Am., 26:26-39.Honji, H. (1981): Streaked flow around an oscillating circular cylinder. J. Fluid Mech., 107:509-520.Howell, J . F . and Novak, M. (1979): Vortex shedding from a circular cylinder in turbulent flow. Proc. 5th Int. Conf. on W i n d Engrg., Paper V - l l .Jacobsen, V., Bryndum, M.B. and Freds0e, J. (1984): Determination of flow kine- matics close to marine pipelines and their use in stability calculations. In Proc. 16th Annual Offshore Technology Conf. Paper O T C 4833.Justesen, P. (1989): Hydrodynamic forces on large cylinders in oscillatory flow. J. Waterway, Port, Coastal and Ocean Engineering, ASCE, 115(4):497-514.Kozakiewicz, A., Sumer, B.M. and Freds0e, J. (1992): Spanwise correlation on a vibrating cylinder near a wall in oscillatory flows. J. Fluids and Structures, 6:371-392.Maull, D.J. and Milliner, M.C. (1978): Sinusoidal flow past a circular cylinder. Coastal Engineering, 2:149-168.Novak, M. and Tanaka, H. (1977): Pressure correlations on a vibrating cylin- der. Proc. 4 t h Int. Conf. on W i n d Effects on Buildings a n d Structures, Heathrow, U.K., Cambridge Univ. Press, pp. 227-232.Obasaju, E.D., Bearman, P.W. a n d G r a h a m , J.M.R. (1988): A study of forces, circulation and vortex patterns around a circular cylinder in oscillating flow. J. Fluid Mech., 196:467-494.
    • 122 Chapter 3: Flow around a cylinder in oscillatory flowsRamberg, S.E. and Griffin, O.M. (1976): Velocity correlation and vortex spacing in t h e wake of a vibrating cable. Trans. ASME, J. Fluids Eng., 98:10-18.Raven, P.W.C., Stuart, R.J. and Littlejohns, P.S. (1985): Full-scale dynamic test- ing of submarine pipeline spans. 17th Annual Offshore Technology Conf., Houston, T X , May 6-9, Paper 5005, 3:395-404.Sarpkaya, T. (1976a): In-line and transverse forces on smooth and sand-roughened cylinders in oscillatory flow at high Reynolds numbers. Naval Postgraduate School, Monterey, CA, Tech. Rep. NPS-69SL76062.Sarpkaya, T. (1986a): Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 165:61-71.Schlichting, H. (1979): Boundary-Layer Theory. 7. ed., McGraw-Hill Book Co.Singh, S. (1979): Forces on bodies in oscillatory flow. P h . D . thesis, Univ. London.Sumer, B.M. and Freds0e, J. (1988): Transverse vibrations of an elastically mounted cylinder exposed to an oscillating flow. Jour. Offshore Mechanics and Arctic Engineering, ASME, 110:387-394.Sumer, B.M., Jensen, B.L. and Freds0e, J. (1991): Effect of a plane boundary on oscillatory flow around a circular cylinder. J. Fluid Mech., 225:271-300.Sumer, B.M., Freds0e, J. and Jensen, K. (1994): A note on spanwise correlation on a freely vibrating cylinder in oscillatory flow. Jour. Fluids and Structures, 8:231-238.Sumer, B.M., Laursen, T.S. and Freds0e, J. (1993): Wave b o u n d a r y layers in a convergent tunnel. Coastal Engineering, 20:3/4:317-342.Tatsumo, M. and Bearman, P.W. (1990): A visual study of the flow around an oscillating circular cylinder at low Keulegan-Carpenter numbers and low Stokes numbers. J. Fluid Mech., 211:157-182.Toebes, G.H. (1969): T h e unsteady flow and wake near an oscillating cylinder. ASME, Journal of Basic Engineering, 91:493-502.Wang, C.Y. (1968): On high-frequency oscillatory viscous flows. J. Fluid Mech., 32:55-68.Williamson, C.H.K. (1985): Sinusoidal flow relative to circular cylinders. J. Fluid Mech., Vol. 155, p p . 141-174.Williamson, C.H.K. and Roshko, A. (1988): Vortex formation in t h e wake of an oscillating cylinder. Jour, of Fluids and Structures, 2:355-381.
    • Chapter 4. Forces on a cylinder in regu- lar waves Similar to steady currents, a cylinder subject to an oscillatory flow mayexperience two kinds offerees: the in-line force and the lift force (Fig. 4.1). In thefollowing, first, t h e in-line force on a smooth, circular cylinder will be consideredand subsequently the attention will be directed to the lift force. T h e remainder ofthe chapter will focus on t h e influence on the force components of t h e followingeffects: surface roughness, angle of attack, co-existing current and orbital motion(real waves).4.1 In-line force in oscillatory flow In steady currents, the force acting on a cylinder in the in-line direction isgiven by F=l-pCDDUU (4.1)where F is t h e force per unit length of t h e cylinder a n d Co is t h e drag coefficient.(Note t h a t t h e velocity-squared term in Eq. 2.8, namely U2, is written in thepresent context in the form of UU to ensure t h a t the drag force is always in thedirection of velocity). In t h e case of oscillatory flows, however, there will be twoadditional contributions to the total in-line force:
    • 124 Chapter : Forces on a cylinder in regular waves Oscillatory flow U = U m sin(cot) ., F L (Lift force) IF (In-Line force) Figure 4.1 Definition sketch. F=^pCDDUU + m U+pVU (4.2) • •in which m1 U is called the hydrodynamic-mass force while pV U is called theFroude-Krylov force where m is the hydrodynamic mass and V is the volumeof the cylinder, which for a unit length of the cylinder reduces to A, the cross-sectional area of t h e cylinder. T h e following paragraphs give a detailed account ofthese two forces.4.1.1 Hydrodynamic mass T h e hydrodynamic mass can be illustrated by reference to the followingexample. Suppose t h a t a thin, infinitely long plate with the width b is immersedin still water and t h a t it is impulsively moved from rest (Fig. 4.2). W h e n the plateis moved in its own plane, it will experience almost no resistance, considering thatthe frictional effects are negligible due to the very small thickness of the plate.Whereas, when it is moved in a direction perpendicular to its plane, there will bea tremendous resistance against the movement. T h e reason why this resistance is so large is t h a t it is not only the platebut also the fluid in the immediate neighbourhood of t h e plate, which has to beaccelerated in this case due to t h e pressure from the plate. T h e hydrodynamic mass is defined as the mass of the fluid around the bodywhich is accelerated with the movement of the body due to the action of pressure.
    • In-line force in oscillatory flow 125 a) m = 0 2 b) m = | b Figure 4.2 Movement of an infinitely long plate in an otherwise still fluid, a) Movement of the plate in its own plane and b) that in a direction perpendicular to its own plane.If the hydrodynamic mass is denoted by m , the force to accelerate the total mass,namely the mass of the body, m, and the hydrodynamic mass, m , may be writtenas F = (m + m)a (4.3)where a is the acceleration. Usually, the hydrodynamic mass is calculated by neglecting frictional effects,i.e. the flow is calculated by expressing fluid force equilibrium between pressureand inertia. Hereby the flow field introduced by accelerating t h e body throughthe fluid can be calculated using potential flow theory. T h e procedure to calculate the hydrodynamic mass for a b o d y placed ina still water can now be summarized as follows. 1) Accelerate the b o d y in thewater; (this acceleration will create a pressure gradient around the body resultingin the hydrodynamic-mass force); 2) calculate the flow field around the body; 3)calculate the pressure on the surface of the body based on the flow information inthe previous step; and finally 4) determine the force on t h e body from the pressureinformation. In the following we shall implement this procedure to calculate thehydrodynamic mass for a free circular cylinder.
    • 126 Chapter J,: Forces on a cylinder in regular wavesExample 4.1: H y d r o d y n a m i c m a s s for a circular c y l i n d e r Figure 4.3 Potential flow around an accelerated cylinder, moving with ve- locity U in an otherwise still fluid. W h e n a cylinder is held stationary and the fluid moves with a velocity Uin the negative direction of the i-axis, the velocity potential is given by (Milne-Thomson, 1962, Section 6.22): </> = U{r + ^)cos9 (4.4) If we superimpose on t h e whole system a velocity U in the positive directionof the x-axis, the cylinder will move forward with velocity U and the fluid will beat rest at infinity (Fig. 4.3), so t h a t <j> is given by (Milne-Thomson, 1962, Section9.20): r2 4> = U-±cos6 (4.5) r T h e velocity components vT and v$ will then be calculated as follows ve = (4.6) r ad rz Vr = U r-I cos0 z (4.7) dr T h e pressure around the cylinder can now be calculated, employing thegeneral Bernoulli equation (Milne-Thomson, 1962, Section 3.60):
    • In-line force in oscillatory flow 127 V 1 9 96 ,. „. - + -v ~ s- = constant (4.8) p 2 atin which v is the speed v2 = v2r + vj (4.9)On the cylinder surface v2 will be u2 = i7 2 (sin 2 0 + cos2<?) = t/ 2 (4.10)therefore the pressure on the cylinder surface from Eq. 4.8 can be written as v dd> , - = - f + constant y(4.11) J p dtwhere the constant term includes also ^U2, as the latter does not vary with theindependent variables r and 6. This term, as a matter of fact, is not significantas it does not contribute to the resulting force. So, dropping the constant, thepressure on the cylinder surface may be written as d<t> O( r x dU p = proa cos 8 (4.12)in which a is the acceleration, i.e. a = dU/dt. The resultant force can then be calculated by integrating the pressurearound the cylinder /•27T P = - / pcose(r0d6) (4.13) JoThe vertical component of the force will be automatically zero due to symmetry.So the resultant force will be /•2JT P = -aprl „2 /I / „„„2, cos 2 6 d6 ./o P = -prlair (4.14) In other words, the force required to accelerate a cylinder with an accel-eration a in an otherwise still fluid should be given by F = ma + pr-wa = (m + m)a (4-15)
    • 128 Chapter 4- Forces on a cylinder in regular wavesand therefore the hydrodynamic mass of a circular cylinder will be given by m = pirrg (4.16) Traditionally, the hydrodynamic mass is written as m = pCmA (4.17)in which A is the cross-sectional area of t h e body (A = Kr for the precedingexample) and the coefficient Cm is called t h e hydrodynamic-mass coefficient. Cmfor a circular cylinder is (Eq. 4.16): Cm = 1 (4.18) Appendix II lists the values of the hydrodynamic-mass coefficients for var-ious two- and three-dimensional bodies.E x a m p l e 4.2: H y d r o d y n a m i c m a s s for a circular c y l i n d e r n e a r a w a l l W h e n the cylinder is placed near a wall (the pipeline problem), the hy-drodynamic mass will obviously be influenced by the close proximity of the wall.Yamamoto et al. (1974) has developed a potential flow solution to account for thiseffect. Their result is reproduced in Fig. 4.4. As is seen, t h e hydrodynamic-masscoefficient Cm increases with decreasing t h e gap between the cylinder and thebed. It is further seen t h a t Cm goes to unity, its asymptotic value, as e/D —• oo.Yamamoto et al. noted that considerations were given for flows accelerating b o t hperpendicular and parallel to the wall; it was found t h a t Cm determined from thetheory was the same regardless of the flow direction. Finally, it m a y be mentioned that simple algorithms for calculating hydro-dynamic mass for cylinders placed near an arbitrarily shaped scoured sea bedwere given by Hansen (1990). Hansens calculations cover also groups of cylinders.A number of examples including multiple riser configurations were given also inJacobsen and Hansen (1990).
    • In-line force in oscillatory flow 129 J I I L 3- e 2.29- 2- 1 1 1 1 0 0.5 1 e/D Figure 4.4 Hydrodynamic-mass coefficient for a circular cylinder near a wall. Yamamoto et al. (1974).4.1.2 T h e F r o u d e - K r y l o v force As seen in the previous section, when a body is moved with an accelerationa in still water, there will be a force on t h e body, namely t h e hydrodynamic-mass force. This force is caused by the acceleration of the fluid in the immediatesurroundings of the body. W h e n the body is held stationary and the water ismoved with an acceleration a, however, there will be two effects. First, the waterwill be accelerated in t h e immediate neighbourhood of t h e body in the same wayas in the previous analysis. Therefore, the previously mentioned hydrodynamicmass will be present. T h e second effect will be t h a t the accelerated motion of thefluid in the outer-flow region will generate a pressure gradient according to dU_ 5 (4.19) dx dtwhere U is t h e velocity far from t h e cylinder. This pressure gradient in t u r n willproduce an additional force on the cylinder, which is termed t h e F r o u d e - K r y l o vforce. T h e force on the body due to this pressure gradient can be calculated bythe following integration:
    • ISO Chapter J: Forces on a cylinder in regular waves = - J pdS (4.20)where S is the surface of t h e body. Prom the Gauss theorem, Eq. 4.20 can bewritten as a volume integral F, = -v%W (4.21)Using t h a t the pressure gradient is constant and given by Eq. 4.19 this gives Fp = pV U (4.22)in which U isFor a cylinder with the cross-sectional area A and with unit length, Fp will be F„ = pA U (4.24)For a sphere with diameter D, on the other hand, Fp will be /7T£>3 • Ff = P U (4 25) {~l~) In t h e case when t h e b o d y moves in an otherwise still water, there willbe no pressure gradient created by t h e acceleration of the outer flow (Eq. 4.19),therefore the Froude-Krylov force will not exist in this case.4.1.3 The Morison equation Now t h e total in-line force can be formulated for an accelerated water envi-ronment where the cylinder is held stationary. T h e total force, F, is given b y Eq.4.2 with the hydrodynamic-mass force given by Eq. 4.17 and the Froude-Krylovforce by Eq. 4.24. Therefore F will be written as F = ]-pCDDUU + pCmA U +pA U (4.26) Drag Hydro- Froude- force dynamic Krylov mass force force
    • In-line force in oscillatory flow 131 T h e preceding equation can be written in t h e following form l F = -PCDDU | U | +P(Cm + 1)A U (4.27) By denning a new coefficient, CM, by CM = Cm + 1 (4.28)Eq. 4.27 will read as follows F = ]-pCDDUU + pCMA U (4.29)This equation is known as the Morison equation (Morison, OBrien, Johnson andSchaaf, 1950). T h e new force term, PCMA U, is called t h e inertia force and the new coeffi-cient CM is called t h e inertia coefficient. (In t h e case of a circular cylinder exposedto an oscillatory flow with small KC numbers such as 0 ( 1 ) , CM{— Cm + 1), tendsto the value 2, since the flow is unseparated in this case (Fig. 3.15) and thereforethe potential-flow value of Cm, namely Cm = 1 (Eq. 4.18), can be used). In the case when t h e b o d y moves relative to the flow in the in-line direction(this may occur, for example, when the body is flexibly mounted) t h e Morisonequation, from Eq. 4.26, can be written as F = ^pCDD(U -Ub)U -Uh +pCmA(u - Ub) + pAU (4.30) Drag Hydro- Froude- force dynamic Krylov mass force forcewhere Ub is the velocity of the b o d y in the in-line direction. Clearly, the Froude- • • •Krylov force must be based on U rather t h a n (U — Ub), because this force isassociated with the absolute motion of the fluid rather t h a n the motion of thefluid relative to the body.T h e d r a g force Fp v e r s u s t h e i n e r t i a force Fi From Eq. 4.29, it is seen t h a t there is a 90° phase difference betweenthe m a x i m u m value of FD and the maximum value of Fi, which is schematicallyillustrated in Fig. 4.5. This phase difference should be taken into consideration ifthe maximum value of the in-line force is of interest. T h e ratio between the m a x i m u m values of the two forces, on the other hand,can be written from Eq. 4.29 as
    • 1SZ Chapter 4- Forces ore a cylinder in regular waves UlUl " Figure 4.5 Time variation regarding the drag- and the inertia force in oscil- latory flows. *>,. CMjD2^Um 2 D CM _ K2 CM (4.31) FD,n CDDUI UmT CD KC CD For small KC numbers, the inertia coefficient CM c a n D e taken as CM — 2,as mentioned in t h e preceding section. Therefore, the force ratio in the precedingequation, taking Cp — 1, becomes •Fj.max _ 20 (4.32) FD,m,K ~ KC
    • In-line force in oscillatory flow 1SS This means t h a t , for small KC values, t h e inertia component of the in-lineforce is large compared with t h e drag component, thus in such cases t h e drag canbe neglected. However, as t h e KC number is increased, t h e separation begins tooccur (Fig. 3.15), and therefore t h e drag force becomes increasingly important.As a rough guide we may consider the range of the Keulegan-Carpenter number0 < KC <C 20 — 30 as the inertia-dominated regime, while KC > 20 — 30 as thedrag-dominated regime. Finally, it may be mentioned t h a t , in some cases such as in t h e calculationof damping forces for resonant structural vibrations, t h e drag force becomes soimportant t h a t even the small contribution to t h e total force must be taken intoconsideration.4.1.4 I n - l i n e force coefficientsExample 4.3: Asymptotic theory For very small KC numbers (such as KC < 1 ) combined with sufficientlylarge Re numbers (such as Re ~ O ( l ) or larger, b u t not too large for the bound-ary layer to be in turbulent regime), it is possible for the case of non-separatingflow to develop an asymptotic theory for determining t h e in-line force coefficients(Bearman, Downie, G r a h a m and Obasaju, 1985b). T h e procedure used in thisasymptotic theory is as follows: 1) Calculate the in-line force on t h e cylinder dueto the oscillating flow, using the potential-flow theory; 2) calculate the oscillating,laminar b o u n d a r y layer over the surface of t h e cylinder; 3) determine the per-t u r b a t i o n to t h e outer flow caused by t h e predicted oscillating laminar boundarylayer; and finally 4) calculate the in-line force on t h e cylinder induced by thisperturbation. This together with the potential-flow in-line force (in Step 1) willbe t h e total in-line force on the cylinder.1) Potential-flow solution: This can be obtained by solving Laplaces equation. Let the resulting solu-tion be Wo(z) where W0(z) is the complex potential, defined by W0(z) = </> + i4> (4.33)in which <j> is the potential function, i/> is t h e stream function and z is t h e complexcoordinate (Fig. 4.6) z = x + iy = re$ (4.34)
    • 134 Chapter 4: Forces on a cylinder in regular waves qeo=qe0e U = Ume Figure 4.6 Definition sketch for potential-flow solution. Boundary layer Figure 4.7 Definition sketch for the boundary layer developing on the cylin- der surface.(Milne-Thomson, 1962, Section 6.0). In t h e case of a circular cylinder, WQ(z) isgiven by 2 W0(z) = U(t)(z + ^fj (4.35)(Milne-Thomson, 1962, Section 6.22), and the velocity U(t) in the preceding equa-tion for the present case is given by U(t) = Ume tut (4.36)T h e in-line force on the cylinder due to this flow can be calculated, using theBlasius formula (Milne-Thomson, 1962, Section 6.41):
    • In-line force in oscillatory flow 1S5 F0 = -ip^ j W0(z)dz (4.37) sInserting Eq. 4.35 in the preceding equation, the force due to this potential flowis obtained as F0 = 2pA U (4.38) 2in which A = nr Since U = Umeiut, then F0 will be F0 = 2pAUmiu>eiut (4.39) Perturbation due to the boundary layer: T h e speed due to the potential flow is calculated by 9o Vu2 + v2 = dW0/dz (4.40)Let qeo be the speed on the surface S of t h e body (Fig. 4.6). From Eqs. 4.35, 4.36and 4.40, qeo is found as follows qeo = | dW0/dzs = <Zeoeiu" (4.41)in which qeo, the amplitude of qeo, is qeo = 2Umsm6 (4.42) In response to the velocity qeo, an oscillatory boundary layer will developeon S (Fig. 4.7). In the case when KC <C 1, and Re ~ 0 ( 1 ) or larger (so t h a t theflow can be represented by an outer potential flow and an inner laminar boundary-layer flow), the boundary layer can be approximated to t h a t which occurs on aplane wall. T h e velocity in such a b o u n d a r y layer is given as (Batchelor, 1967, p.354) 31=51^ (4.43)in which <7i=<?e„(l-e-(1+)an) (4.44)Here, a is « = ( - ) (4.45)
    • 1S6 Chapter J,: Forces on a cylinder in regular wavesand n being the local coordinate (Fig. 4.7) measured normal to the surface S ofthe body. This b o u n d a r y layer will p e r t u r b the previously predicted potential-flowforce in the following two ways: 1) T h e wall shear stress caused by the boundarylayer will contribute to t h e force (the friction force); and 2) t h e growth of theboundary layer will p e r t u r b the outer flow, and this will in t u r n p e r t u r b thepressure on the surface of the body, resulting in an additional contribution to theforce.The friction force: T h e in-line component of t h e force due to t h e wall shear stress on S (thefriction force) is 2TT Ff= J rwsm6ds (4.46) 9=0in which dqi = /i(l + i)aqeo (4.47) onand s being the local coordinate (Fig. 4.7) measured along t h e surface S in t h edirection of 6. Inserting Eq. 4.47 into Eq. 4.46, and using Eqs. 4.41 and 4.42, Ffis obtained as follows Ff = i ( l + i)^D2UmQ1/2e^ (4.48)in which 1 D2uj Re u AO ^ (4 48a) -,^T = KC -The force due to pressure perturbation T h e growth of the b o u n d a r y layer is not uniform over the surface S of thebody. If S* is the displacement thickness of the b o u n d a r y layer, oo 6= / 7 i - i L ) d n = ^ _ , (4.49) J qeoJ (l+i)a othe product qeoS* will represent the flux deficit at section s (Fig. 4.8). T h equantity Jj(ge0<S*)d.s will then represent the difference between the flux deficitsat sections s and s + ds. This fluid, namely -^{qeo8*)ds, is entered into the outer
    • In-line force in oscillatory flow 1S7 3 7 (qe 0 S*)ds Boundary layer Figure 4.8 Fluid entrainment into the outer potential flow due to growing boundary layer.potential flow over t h e length ds (this is the perturbation caused by the boundarylayer). T h e aforementioned effect can be considered as a source with the strengthm determined from t h e following equation (see Milne-Thomson, 1962, Sections8.10 and 8.12 for source and its complex potential) 2 — [qeoS*)ds = 27rm ds (4.50) os and the corresponding complex potential function can be written as Wi(z) = <f) — m l o g ( z — z(s))ds s -^hVafd-tl0&{Z-Z{S))dS (451) This complex potential will create an additional pressure on the surface Sof the body, and t h e force caused by this additional pressure can be calculated bythe Blasius formula Fp = -ipjt iWxWdz - pjt I Im{Wx(z)dz (4.52) swhere the second integral represents the contribution from the fact t h a t the streamfunction of t h e complex potential, namely Im{W(z)}, is not a constant on 5 .Using Eq. 4.51, t h e above integrals were calculated analytically by Bearman et al.(1985b) and the result is
    • 138 Chapter Jft Forces on a cylinder in regular waves F P = 0-+ *)P"D2Um Q 1/2 eiwf (4.53) As seen from Eqs. 4.48 and 4.53, the friction force and the pressure forceapparently are equal.3) Total in-line force and in-line force coefficients T h e total in-line force is obtained from Eqs. 4.39, 4.48 and 4.53 as F = F0 + Ff + Fp = = 2pAUmiui eiwt+ + (l + i)pwD2UTnQy/2el"t (4.54)T h e same force due t o the Morison formulation is F=^pCDDUU+pCMAU (4.55)Inserting Eq. 4.36 in Eq. 4.55 and making the approximation t h a t , over a flowcycle, eiwteiut ~ (8/(37r))e ! u ", t h e Morison force can be written as + pCMAUmuieiujt (4.56)From Eqs. 4.54 and 4.56, the in-line force coefficients are found as follows <7M = 2 + 4 ( 7 T / ? ) - 1 / 2 (4.57) - 1 2 CD = ^ ( t f C r V / ? ) (4-58) Stokes (1851) was the first t o develop an analytical solution for the case ofa cylindrical body oscillating sinusoidally in a viscous fluid. His solution is givenin the form of a series expansion in powers of (Re/KC)-12. T h e results of theasymptotic theory given in the preceding paragraphs are t h e same as the Stokesresults to 0 [ ( i ? e / A " C ) _ 1 / 2 ] . Subsequently, Wang (1968) extended Stokes analysisto 0 [ ( f l e / 7 i C ) _ 3 / 2 ] , implementing the method of inner and outer expansions. Fig. 4.9 compares the results of the asymptotic theory with those of exper-iments by Sarpkaya (1986a) for the value of the /?(= Re/KC) p a r a m e t e r of 1035.As is seen, the theory shows remarkable agreement with the experiments for verysmall values of KC where the flow remains attached (cf. Fig. 3.15).
    • In-line force in oscillatory flow 139 P (= Re/KC) = 1035 2.4 2.0 <*** 1.6 1.4 1.2 1.0 0.8 0.6 Asymptotic theory 0.4 _l I L 0.2 0.4 l.O 2.0 4.0 10.0 20.0 KC Asymptotic theory 2.4 2.0 1.6 1.4 1.2 1.0 % 8 0.8 0.6 0.4 _L _i i i 0.2 0.4 1.0 2.0 4.0 10.0 20.0 KCFigure 4.9 Drag and inertia coefficients vs Keulegan-Carpenter number. Re/KC = 1035. Experiments from Sarpkaya (1986a). Asymp- totic theory (Eqs. 4.58 and 4.57).
    • HO Chapter 4- Forces on a cylinder in regular wavesM e a s u r e m e n t s o f Cry a n d CM coefficients T h e preceding analysis indicates t h a t the in-line cofficients are dependenton two independent variables, namely the Reynolds number and the Keulegan-Carpenter number. T h e theory gives the explicit form of this dependence. How-ever, this is for t h e combination of very small KC numbers and sufficiently largeRe numbers only. Although there are several numerical codes developed to calcu-late flow around and forces on a cylinder in oscillatory flows (Chapter 5), these arestill at the development stage and therefore not fully able to document the way inwhich the force coefficients vary with KC and Re. Hence, t h e experiments appearto be the most reliable source of information with regard to the force coefficientsat the present time. There are various techniques to determine the coefficients Cu and CM e x _perimentally. For periodic flows, the most suitable technique may be " t h e methodof least squares". T h e principle idea of this method is t h a t the Co and CM coeffi-cients are determined in such a way t h a t the mean-squared difference between thepredicted (by t h e Morison formula) and t h e measured force is minimum. A briefdescription of the method of least squares is given below. Let Fm(t) be the measured in-line force at any instant t. Likewise, let Fp(t)be the predicted in-line force corresponding to the same instant, namely 1 FP(t) = -pCDDU(t)U(t) + PCMA U (t) (4.59) 2Let, for convenience: fd = ^pCDD and fi = PCMA (4.60a, b) Therefore, the predicted force: Fp(t) = fdU(t)U(t) + f,U (t) (4.61) 2 Now, let £ be the sum of the difference between the predicted force andthe measured force over the total length of the record: e* = J2[Ff(t)-Frn(t)}2 = E [hU{t)U{i) + f,U (t) - Fm(t)}2 (4.62)For e 2 to be minimum: fir2 de2 dfd dfiT h e first equation leads to:
    • In-line force in oscillatory flow 141 / ( E ^ w ) +/<(Et/wi[/wi £(*)) = E^)i^(*)i^m(o (4.64)and the second equation leads to: /„ ( £ U(t)U(t) U (0) + fi ( E U (*)) = E ^ (0^(0 (4-65)where the summation is taken over the total record length. Eqs. 4.64 and 4.65form two simultaneous equations with fi and / ; as unknowns. Solving for fd andfi, the in-line force coefficients Co and CM can b e determined from Eqs. 4.60aand b , respectively. For a sinusoidal flow, it can be shown t h a t the method of least squares givesCD and CM as follows: F °D = V-~i^7Y I ™ cos(w*) I cos(wi) I d(Lot) (4.66) 1KC 1 f2lr CM = j ^777/ D Fmsn(ut)d{u*) (4.67) * P U^ Jo Given the time series of the measured force Fm{t), the Co and CM coeffi-cients can therefore be worked out from the preceding equations. Another technique regarding the experimental determination of Co and CMcoefficients is the Fourier analysis. This latter technique yields identical CM values.As for Co, the Co values obtained by the Fourier analysis differ only slightly fromthose obtained by t h e method of least squares (Sarpkaya and Isaacson, 1981). Keulegan and Carpenter (1958) were the first to determine t h e Co andCM coefficients for a cylinder exposed to real waves (using the Fourier analysis).Subsequently, Sarpkaya (1976a) m a d e an extensive s t u d y of t h e forces on cylindersexposed to sinusoidally varying oscillatory flows (created in an oscillatory U-shapedtube) with the purpose of determining the force coefficients in a systematic manneras functions of t h e Keulegan-Carpenter number and t h e Reynolds number as wellas the relative roughness of the cylinder.V a r i a t i o n o f Co a n d CM w i t h KC number T h e variation of Co and CM with KC has already been illustrated in Fig.4.9 in conjunction with t h e asymptotic theory (Example 4.3). T h e range of KCcovered in the figure was rather small. Fig. 4.10 illustrates this variation, coveringa much wider range of KC number up to about 60. T h e Reynolds number for the
    • 142 Chapter 4- Forces on a cylinder in regular waves • • • l _ _ l • • • • • • • Re = 1.7 x 10 2- ^-"^ Asymptotic theory 1 "I— I I I I I I I "I" I I I I 1 I I—I I I I I 0.03 0.1 10 100 • • • • • I I I I t I 111 I . I t I . I 11 3- Asymptotic theory 2- 1- -I 1—n-TTTT -| 1—1 I 1 I I j I 0.03 0.1 10 100 KC Figure 4.10 Variation of in-line force coefficients with KC number for a given Re number, namely Re = 1.7 X 10 4 . Data from X: Sarpkaya (1976a), o, a: Bearman et al. (1985a), and A : Anatiirk (1991). Asymptotic theory (Eqs. 4.57 and 4.58).d a t a given in t h e figure is constant (Re = 1.7 x 10 4 ). T h e results of the asymptotictheory for the same Re number are also included in the figure. First consider the drag coefficient. As seen from the figure, there are threedistinct regimes in the variation of Co with KC: 1) KC ~ 0.3, 2) 0.3 ~ KC ~ 13and 3) KC ~ 13.
    • In-line force in oscillatory flow 14S In t h e first regime, namely KC < 0.3, the drag coefficient must be governedby t h e asymptotic theory summarized in Example 4.3, as the conditions for theapplication of the asymptotic theory are fully satisfied, namely KC is very small,Re is sufficiently large, and the flow remains attached (Fig. 3.15). Unfortunately,no experimental d a t a exist in the literature for this particular Re number in thisrange of KC to confirm t h e validity of t h e application of t h e asymptotic theory. W h e n KC = 0.3 is reached, separation begins to occur. Therefore, the dragwill no longer be governed by t h e asymptotic theory. Hence, the Co variation willbegin to diverge from the line representing the asymptotic theory in Pig. 4.10. T h efigure indicates t h a t this regime of Cry variation with KC extends up to KC ~ 13.Apparently, this latter value of KC coincides with t h a t corresponding to the upperboundary of the transverse-vortex-street regime described in Section 3.2. W h e nKC is increased beyond KC ~ 13, the transverse vortex street will disappear,and the shed vortices will form a vortex street lying parallel to the direction ofthe oscillatory motion, in much the same way as in steady current. Therefore thedrag coefficient will in this regime {KC > 13) not change very extensively withKC. Regarding the inertia coefficient, CM, from Fig. 4.10, here, too, there arethree different regimes, namely: 1) KC ~ 6, 2) 6 ~ KC ~ 13 and 3) KC ~ 13,the boundary between the first two regimes, namely KC = 6, being different,however, from t h a t corresponding to the drag coefficient Co- As for t h e first regime, KC < 6, the asymptotic theory predicts the CMcoefficient extremely well. However, when KC reaches the value of approximately6, an abrupt fall occurs in CM (the so-called inertia crisis). This a b r u p t fallcontinues over the range from KC = 6 to 13. KC ~ 6 coincides with the lower limit of the vortex-shedding regimes (Sec-tion 3.2). T h e rapid change in CM a t this value of KC number may therefore bea t t r i b u t e d to t h e vortex shedding. T h e interaction between the vortex sheddingand the hydrodynamic process generating the hydrodynamic mass may producethis observed, sudden drop in the CM coefficient. T h e reduction in CM is so larget h a t , subtracting the Froude-Krylov part of CM, namely unity, from t h e measuredvalues of CM, it is found t h a t the inertia cofRcient (Cm = CM — 1) will take evennegative values for KC values around KC = 10, as seen from Fig. 4.10. As for the third regime in the variation of CM with KC, namely the rangeKC ~ 13, the vortex street formed by the shed vortices in this range lies parallelto the direction of t h e oscillatory motion, as mentioned previously. Therefore thechange in Cm (or CM) with increasing KC in this range will not be very extensive.Effect o f Re n u m b e r o n Co a n d CM Fig. 4.11 presents the in-line-force-coefficient data, illustrating the effect ofRe. T h e drag coefficient diagram includes also CQ versus Re variation for steadycurrents (Fig. 2.7) to facilitate comparison. T h e figure is based on the results
    • 144 Chapter 4- Forces on a cylinder in regular waves Sarpkaya(1976) Extended curves based on the following data KC: 6 8 10 15 * 0 a o Justesen(1989) ( B S D Sarpkaya(1986a) 2.0 ••• in- 1 11 1 1 1 i i i i i 111 i I I I - 20--:.. ••. 1.8 - -.8 6GX. 1.6 : """-"^ ••-.6 ••. " 1.4 100 1.2 - - •-. ••-$?•:•, ^ ^ 10 1.0 ^ = » = 8 = 15 : 0.8 * B- s ; - 2 ^ -—?.?TT.7r......... Steady " 0.6 -r.lo-.--- "qS6 current - i 0.4 i i i i i i i i i 11 i 1 T 0.2 10 10 10 Re 2.0 -LJ^I 1 1 i1 i 1 1 1 1 1 II 1 i 1 1 1 1 40 60 100 1.8 8 1« ^v> v20 ":-~ * ••••"::>-":::::---vci : . . . . . - • •"••-•••••• •«• 1.6 V*-C».-..-=--»-_*"-- e 8 40 ^ K " " 00 ^-"-^—r-^"> 1.4 100 s ~°--o-- 15 - - ^ 1.2 > - 1.0 - 40 . 2 . Q . •;.:.•••• ., _ •"••••ID ..-• 0.8 " 1.5--" i i i 0.6 • i i i i i i i i i I I I 10 10 10 Re Figure 4.11 In-line force coefficients for a free, smooth cylinder. Steady cur- rent CD variation is reproduced from Fig. 2.7 which is originally taken from Schewe (1983). Oscillatory flow data are from Sarp- kaya (1976a), Sarpkaya (1986a) and Justesen (1989).
    • In-line force in oscillatory flow 145of the extensive study of Sarpkaya (1976a and 1986a) and t h e study of Justesen(1989). It is apparent from t h e figure t h a t the drag coefficient varies with Re inthe same m a n n e r as in steady currents. However, the drop in Cr) with Re (whichis known as t h e drag crisis in steady currents, see Section 2.2) does not occur asabruptly as in steady currents. For a given KC number, Cp first experiences a gradual drop with increas-ing Re number. Similar to the steady currents, this range of Re number may beinterpreted as the lower transition regime (see Section 2.2). Subsequently a rangeof Re number is reached where Cp remains approximately constant. This maybe interpreted as the supercritical Re-number regime. Following t h a t , CD beginsto increase with an increase in Re, interpreted as the upper transition .Re-numberregime. Finally, the Cp coefficient reaches a plateau where it remains approxi-mately constant with increasing Re. This latter regime, on t h e other hand, maybe interpreted as t h e transcritical .Re-number regime. Regarding the inertia coefficient in Fig. 4.11, the general trend is oppositeto t h a t observed for Cp. W h e r e Cp experiences high values, CM experiences lowones. T h e increase in CM may be due to the weak vortex-shedding regime whichtakes place in the supercritical flow regime and particularly in t h e upper-transitionflow regime.Example 4.4: Effect of friction o n CD a n d CM In C h a p t e r 2, based on the experimental d a t a obtained for steady currents,it was demonstrated t h a t , for most of the practical cases, the friction drag is onlya small fraction of the total drag (Fig. 2.4). Regarding t h e oscillatory flows, unfortunately no d a t a are available in t h eexisting literature, therefore no conclusion can be drawn with regard to the effectof friction on the in-line force. Nevertheless, this effect may be assessed, utiliz-ing Justesens (1991) theoretical analysis. T h e results depicted in Fig. 4.12 arefrom the work of Justesen (1993, private communication), which is an extensionof Justesen (1991) where a numerical solution was obtained to a stream function-vorticity formulation of the Navier-Stokes equations for t h e flow around a circularcylinder at small KC numbers in the subcritical Reynolds number range. Al-though the results are limited to small Re numbers, they nevertheless illustratethe influence of the friction on the force coefficients. Regarding t h e drag coefficients, Fig. 4.12 indicates t h a t the friction is ex-tremely important for small KC numbers. As a m a t t e r of fact, the contributionof friction to the total drag is 50% for very small KC numbers (KC = 0 ( 1 ) orless), as predicted by t h e asymptotic theory (Example 4.3). As KC is increased,however, the diagram indicates t h a t t h e effect of friction on the drag gradually
    • 146 Chapter 4- Forces on a cylinder in regular waves 10 1 l TTTTTI l — i — r - i i ii i 111 - - - — - " - Total - V ^Ss %. ^s/ ~ / ~ — // — ~ IS -s — - 1 Due to pressure Asymptotic ~ theory ~ 0.1 i i 0.1 10 KC ~i—i—i i i i II i 1 — i — r i i 11 II 1—i—r Total Asymptotic theory i i i i i 1111 i i i i i i i l l L 0.1 10 KC Figure 4.12 Effect of friction on the force coefficients. /?(= Re/KC) = 196. From numerical solution of Navier-Stokes equations in the sub- critical Re number range. Justesen (1993, private communi- cation), which is an extension of Justesen (1991). Asymptotic theory: Eqs. 4.57 and 4.58.decreases; at KC = 6, for example, the friction drag becomes less t h a n 10% of thetotal drag. Therefore, for large KC numbers, the drag portion of t h e in-line forcemay be considered to be due to pressure alone. Regarding the inertia coefficient, on the other hand, it is seen from Fig.4.12 t h a t the friction-generated inertia force is only a very small fraction of the
    • In-line force in oscillatory flow 147 1 F 3 JpDUm 2 Morison y "y>-^. Measured "A y y 1 / y 0 * -1 ^ s // // // N -2 -3 1 1 1 J_. 1 1 _l .._ .1 ~. -90° 0° 90° 180° 270° oit Figure 4.13 Comparison of measured and Morison-predicted in-line force. KC = 14, Re = 2.8 X 10 4 . Sarpkaya and Isaacson (1981).total inertia force (less t h a n 4% at best). Therefore it may be neglected in mostof the practical cases.4.1.5 Goodness-of-fit of t h e Morison equation Fig. 4.13 gives a comparison between the measured and Morison-predictedin-line forces. Clearly, the Morison representation is not extremely satisfactorywith respect to t h e measured variation of the in-line force. T h e question how wellthe Morison equation represents the measured in-line force has been the subjectof several investigations (Sarpkaya and Isaacson, 1981). In order to assess the applicability of the Morison equation, one may intro-duce a goodness-of-fit parameter, S, denned by I(Fm - Fpfdt 8 = ^—57 (4.68) fFldt 0in which Fm and Fp are the measured and the predicted (by Morisons equation)forces, respectively, and Tt is the total duration of d a t a sampling. Fig. 4.14 showsa typical variation of 5 with respect to KC. As is seen, < increases from zero S
    • 148 Chapter ^: Forces on a cylinder in regular wavesfor small KC to a maximum at KC = 12 where 8 attains a value of S = 0.12,and with further increase in KC, 6 decreases again. Clearly, t h e ability of theMorison equation to predict the force depends heavily on the KC number. In theinertia-dominated region, S is rather small, therefore the Morison representation israther good, b u t when the flow is separated, the Morison equation can not providea complete description of the force variation. To tackle this problem Sarpkayaintroduced a four-term Morison equation which may be written as {ir2/KC) CM sin6 - CD cos 0 cos 6 + PDUI A _ 1 / 2 [ 0 . 0 1 + 0.1 exp{-0.08(A"C - 12.5) 2 }] cos[36i- A ~ 1 / 2 ( - 0 . 0 5 - 0 . 3 5 e x p { - 0 . 0 4 ( A C - 12.5) 2 })] + A _ 1 / 2 [ 0 . 0 0 2 5 + 0 . 0 5 3 e x p { - 0 . 0 6 ( A C - 12.5) 2 }] cos[56»- A _ 1 / 2 ( 0 . 2 5 + 0.6exp{-0.02(is:C - 12.5) 2 })] (4.69)in which 6 = ujt and A = (2 - CM)/(KC CD). T h e results have shown t h a t , in this way, a significant improvement hasbeen obtained. (Sarpkaya (1981) and Sarpkaya and Wilson (1984)). 0.15 0.10 0.05 .~3 L 10 15 KC Figure 4.14 Goodness-of-fit parameter S as function of KC. Re = 5 X 10 . Smooth cylinder. Justesen (1989).
    • Lift force in oscillatory flow 1^94.2 Lift force in oscillatory flow- W h e n a cylinder is exposed to an oscillatory flow, it may undergo a lift force(Fig. 4.1). This lift force oscillates at a fundamental frequency different from thefrequency of the oscillatory flow. T h e time variation of the force is directly relatedto the vortex motions around the cylinder, as has already been discussed in Section3.2. Obviously, if the flow around the cylinder is an unseparated flow (very smallKC numbers, Figs. 3.15 and 3.16), then no lift will be generated. Fig. 4.15 illustrates the emergence and subsequently t h e development of thelift force as the KC number is increased from zero. T h e figure indicates t h a t , whilethe lift force first comes into existence when KC becomes 4 (which is due to theasymmetry in the formation of the wake vortices; see Fig. 3.2.e), well-establishedlift-force regimes are formed only after KC is increased to the value of 6-7, thevalue of KC number beyond which vortex shedding is present. W h e n the analysis of the lift force is considered, the most important quan-tities are the fundamental lift frequency and the magnitude of t h e lift force. Regarding the fundamental lift frequency, this has been discussed in detailsin Sections 3.2 and 3.3, and the normalized fundamental lift frequency J V L ( =/ l / / u > ) i namely the number of oscillations in t h e lift per flow cycle, has beengiven in Table 3.1 and in Fig. 3.16. As regards the magnitude of the lift force, there are two approaches. Inone, the maximum value of t h e lift force is considered, while in the other t h e root-mean-square (r.m.s.) value of the lift force is adopted to represent t h e magnitudeof the lift force. These may, in terms of the force coefficients, be written in thefollowing forms: FLM» = pCL^DU2m (4.70)and f i r m , = pCLlmsDUl (4.71)i n w h i c h -Fornax and FiIlns are the maximum- and r.m.s.-values of the lift force,respectively, while Ci, m a , x and Citms are t h e corresponding force coefficients. Ifthe time variation of t h e lift force is approximated by a sinusoidal variation, thenthe two coefficients will be linked by t h e following relation ^Lmax — V 2 C ^ r m s (4.72)
    • 150 Chapter 4: Forces on a cylinder in regular waves U(t) o WWVWW 1 2 3 4 5 6 7 8 Flow regime: No lift * KC= 1 No separation No lift / Honji regime A pair of asymmetric vortices y 0 -2 8 Single-Pair vortex shedding ;.v^^YV/vf#AfAfAf^ 10 12 14 irf^AfAf^fJ^fJfA 18 Double-Pair / vortex shedding F y 0 y^yvj^)U ^v>^wi wvf-<|/f ^ A 20 -2 r^/f ^^^NW^MA^ 26 Three-Pairs vortex shedding t/T Figure 4.15 Computed lift force traces over nine periods of oscillation at var- ious ifC-values for /?(= Re/KC) = 196. Justesen (1991). For the various flow regimes indicated in the figure, see Figs. 3.15 and 3.16.
    • Lift force in oscillatory flow 151 Lrms 1 - 10 15 KC 20Figure 4.16 Variation of r.m.s. lift-force coefficient as function of KC num- ber. Experimental data from Justesen (1989).Figure 4.17 Lift force r.m.s. as function of KC for a given value of Re/KC) = 730. Willi amson (1985).
    • 152 Chapter 4: Forces on a cylinder in regular waves i i i I I I • i - 1 — • r • • I i i i i i 1 - i K C = 10 Lmax 3 - - ~.20 - .30 - 60 ••,X--.:.;,... - ~ Steady current i i i i i i i i 1 p i I 10 10 10 Re Figure 4.18 Maximum lift coefficient for a free, smooth cylinder. Oscillatory flow data from Sarpkaya (1976a). Steady-current Ci variation is reproduced from Fig. 2.8 where (C*f ) 1 / 2 is multiplied by y/2 to get the maximum lift coefficient, assuming that the lift varies sinusoidally with time. Both C i m a x and C^ms are functions of KC and Re. Fig. 4.16 gives CLUUSas a function of KC number for Re = 2.5 x 10 5 (Justesen, 1989). T h e figure indi-cates t h a t the lift force experiences two maxima, one at KC around 10 and a slightmaximum at KC around 16. This behaviour has been observed previously alsoby authors such as Maull and Milliner (1978), Williamson (1985), a n d Sarpkaya(1986b, 1987). In Williamsons (1985) representation, the product CLlms(KC)2(rather t h a n Ciims) has been plotted as a function of KC. This obviously mag-nifies the aforementioned effect significantly. Williamsons diagram is reproducedhere in Fig. 4.17. T h e figure clearly shows t h a t C i r m s attains m a x i m u m values atKC = 11, 18 and 26. Williamson points out t h a t these peaks probably reflect anincrease in the repeatability of the shedding patterns. Each peak corresponds to acertain p a t t e r n of shedding; namely, the first peak corresponds to t h e single-pairregime (7 < KC < 15), the second to the double-pair regime (15 < KC < 24),and the third to the three-pairs regime (24 < KC < 32). Apparently, these peakscoincide with the KC numbers at which large spanwise correlations are measured.T h e minima in t h e diagram, on the other hand, correspond to t h e KC numberswhere the spanwise correlation is measured to be relatively low, cf. Fig. 3.28.As discussed in Section 3.5 in relation to Fig. 3.28, t h e preceding behaviour islinked to the fact t h a t the correlation is measured to be large (and apparentlyCi, r m s experiences m a x i m u m values) at certain KC numbers because these KC
    • Effect of roughness 15Snumbers lie in the centre of t h e corresponding KC regimes, while the correlationis measured to be low (and, as a result, C x r m s experiences minimum values) atcertain KC numbers because these KC numbers lie at the boundaries betweenthe neighbouring KC regimes. Finally, Fig. 4.18 presents the lift-force d a t a , illustrating the effect of Renumber on the lift force. T h e figure includes also the steady current d a t a whichare reproduced from Fig. 2.8 to facilitate comparison. As is seen, the effect of Reis quite dramatic (see the discussion in Section 2.3).4.3 Effect of roughness W h e n t h e cylinder surface is rough, the roughness will affect various aspectsof the flow, such as t h e hydrodynamic instabilities (vortex shedding and interactionof vortices), the separation angle, t h e turbulence level, the correlation length,and the vortex strength. In addition to these effects, it increases t h e cylinderdiameter, and the projected area. Therefore it must be anticipated t h a t the effectof roughness upon the force coefficients can have some influence. Fig. 4.19 shows the influence of roughness on the in-line force coefficients.T h e d a t a come from the work by Justesen (1989). It must be emphasized t h a tthe experimental system in Justesens work was maintained the same for all thethree experiments indicated in the figure, and the experiments were performedunder exactly the same flow conditions. It is only the cylinder roughness whichwas changed. Therefore, the change in the force coefficients is directly related tothe change in the roughness. T h e figure shows t h a t the drag coefficient increases and the inertia coeffi-cient decreases when the cylinder is changed from a smooth cylinder to a rough onewith k/D = 3 x 1 0 - 3 . Furthermore, it is clear t h a t CD increases with increasingroughness. Apparently CM is not influenced much with a further increase in theroughness. Regarding the increase in Co with increasing roughness, this may be in-terpreted in the same way as in steady currents, considering t h a t the Reynoldsnumber of the tests, namely Re = 5 x 10 5 , is in the post-critical range (see Figs.2.11 and 2.14, and also the discussion in Section 2.2). Regarding the decrease in CM, on the other hand, a clear explanation isdifficult to offer. T h e non-linear interaction between the vortex shedding andthe hydrodynamic process generating t h e hydrodynamic mass - the mechanismbehind the reduction in the hydrodynamic mass in the vortex-shedding-regimeKC numbers - must occur more strongly in t h e case of rough cylinder, since thereduction in CM is much larger in this case t h a n in the case of smooth cylinder. Fig. 4.20 illustrates t h e effect of cylinder roughness on Co and CM whenKC is kept constant (KC = 20, in the presented figure), while Re is changed. It
    • 154 Chapter 4- Forces on a cylinder in regular waves _i i < • i _1 1 L_ Re = 5 x 10 k/D: •- 20x10 Smooth 0- A s y m p t o t i c theory 1 •< i ^ i i 0.1 1 10 KC 1 1 1 — 1 1 1 • 1-1 U • i • • • i 1 1 1— Asymptotic theory / k/D: ^ Smooth • -3 ^.20x10 v . -3 v. „—3 x 10 ! , ,—r-i i i i [ T 1 1—1—1—1 11 1 — i i i 0.1 10 KC Figure 4.19 Effect of roughness on in-line force coefficients. Experimental data from Justesen (1989). Asymptotic theory: Eqs. 4.57 and 4.58.
    • Effect of roughness 155 1.9 "" —"^ 1 k/D = 1 J^-r-4 MIT • 1 20 x l O ^ • • . 1.8 _ a) 1.6 J AV^m/*L: 10x 1° 3 S* A A__ A A A , A - / -i A/ A A /5xlO / „ A 1.4 - * - - JK"""*"" * ^•. AfA 2.5x10 A 1.2 - ~ A 7* — - VV /l.25x 10-3/ / A 1.0 -- - - KC = 2 0 - 0.8 - /* — A^-^4 ••.. y SMOOTH " — .... 0.6 1 1 1 1 1 1 1 1 1 r i i i 1 10 103 2.0 ~ r "i r b)M 1.8 Figure 4.20 Effect of roughness on CD and CM versus Re variation. Sarp- kaya (1976a).
    • 156 Chapter J,: Forces on a cylinder in regular wavesis interesting to note that the way in which Co versus Re variation changes withrespect to the roughness is quite similar to that observed in t h e case of steadycurrents (Fig. 2.11). As far as the lift force is concerned, Fig. 4.21 illustrates t h e effect of thechange in roughness on the lift coefficient. Note that the depicted d a t a are fromthe same study as in Fig. 4.19. Again, the effect is there. It appears t h a t the lift generally increases whenthe cylinder is changed from a smooth cylinder to a rough one. Similar resultswere obtained also by Sarpkaya in his work where the parameter /?(= Re/KC)was kept constant while KC was changed (see Sarpkaya (1976a) and Sarpkaya andIsaacson (1981)). 1 1 1 1 5 Re = 5 xlO Lrms k/D: . 20 x 1 0 3 2 - / - . ~~Jf -3 - / ^ 3 x 10 - 1 — _ //J Smooth .•Ki--^^r | | | 0 10 15 20 KC Figure 4.21 Effect of roughness on lift coefficient. Experimental data from Justesen (1989). Finally, it may be noted t h a t the subject has been investigated very exten-sively since the mid seventies. This is among other things because of its importancein practice where the roughness is caused by the marine growth. For further im-plications of t h e effect of surface roughness on t h e force coefficients, t h e readeris referred to the following work: Sarpkaya (1976a, 1977b, 1986b, 1987, 1990),Rodenbusch and Gutierrez (1983), Kashara,Koterayama and Shimazaki (1987),Justesen (1989), Wolfram and Theophanatos (1989), Wolfram, Javidan and Theo-phanatos (1989) and Chaplin (1993a) among others.
    • Effect of coexisting current 1574.4 Effect of coexisting current If current coexists together with waves, the presence of current may affectthe waves. T h e problem of wave-current interaction is an important issue in itsown right. Detailed reviews of the subject are given by Peregrine (1976), Jonsson(1990) a n d Soulsby, H a m m , Klopman, Myrhaug, Simons and T h o m a s (1993). In t h e following discussion, for the sake of simplicity, we shall considert h a t the oscillatory flow, which simulates the waves, remains unchanged in thepresence of a superimposed current. Let Uc be t h e velocity of t h e current. T h ekey parameter of t h e study will therefore be the ratio of the current velocity to themaximum value of the velocity of the oscillatory flow, namely Uc/Um. Althoughthere are several alternatives with regard to the definition of the Reynolds numberand the Keulegan-Carpenter number in the present case, the definitions adoptedin the case of pure oscillatory flow, namely, Re = UmD/u a n d KC UmTw/Dmay be maintained. b) 0 Ue/Uin = 0 U c / U m = 0.5 u c /u m =i , 1 i -—X / N .^ " - - / ,. - 2K 4 it 2K 4 It , 1 i i4pD(U m + Uc) A A ^ / r A, j. r ^ »V n.-~ u ^ c trouh* 1 a , period-*pD(Um + Uc) 0 "(i A n thtx MA tot - ;yvV W M ^ - Vl|>^—] ^ U ~ — - Figure 4.22 Force time series in the case of coexisting current. KC = 20. Sumer et al. (1992).
    • 158 Chapter 4: Forces on a cylinder in regular waves T h e effect of coexisting current on forces has been investigated by severalauthors. These investigations include those by Moe a n d Verley (1980), Sarpkayaand Storm (1985), Justesen, Hansen, Freds0e, B r y n d u m a n d Jacobsen (1987),Bearman a n d Obasaju (1989) and Sumer, Jensen a n d Freds0e (1992). T h e effect of current on forces can be described by reference t o Fig. 4.22.T h e force traces depicted in t h e figure are taken from t h e study of Sumer et al.(1992) where t h e oscillatory flow was generated b y t h e carriage technique, whilethe current was achieved by recirculating water in t h e flume. From t h e figure t h e following observations can be made: 1) T h e in-line force varies with respect to time in t h e same fashion as t h eflow velocity. 2) T h e way in which t h e lift force varies with time during t h e course of oneflow cycle changes markedly as t h e parameter Uc/Um is changed from 0 t o 1. ForUc/Um = 0.5, t h e portion of t h e flow period where t h e flow velocity U < 0 is justlong enough t o accomodate shedding from both t h e upper and t h e lower sides ofthe cylinder; this is characterized by one positive and one negative lift force in t h elift force trace, Fig. 4.22. For Uc/Um = 1, however, t h e figure shows t h a t t h e shedding disappears(which is characterized by t h e non-oscillating portions of t h e lift force traces)when t h e oscillatory component of t h e motion is in t h e direction opposite t o t h ecurrent. 3) During t h e time periods when t h e vortex shedding exists, t h e figureindicates t h a t t h e Strouhal relation 5 ( 7) 43 < = (^rb -is satisfied provided t h a t t h e velocity is taken as t h e s u m of t h e current velocityUc and t h e wave velocity Um • Here / „ is t h e average vortex-shedding frequency. Regarding t h e in-line force coefficients, t h e Morison equation may beadopted in t h e present case in t h e same format as in Eq. 4.29, b u t with t h evelocity U(t) defined now in t h e following way U = Uc + Um sin(urf) (4.74) Fig. 4.23 presents t h e Co a n d CM coefficients as functions of the parameterue/um. The drag coefficient generally decreases with t h e ratio Uc/Um. It ap-proaches, however, t h e asymptotic value (shown with dashed lines in t h e diagram)measured for steady current for t h e same surface roughness a n d t h e same Renumber, as Uc/Um —* oo, as expected. The inertia coefficient, CM is apparently not very sensitive t o Uc/Um exceptfor t h e KC = 5 case. T h e discrepancy between t h e results of Sumer et al.s(1992) study a n d those of Sarpkaya and Storm (1985) m a y be a t t r i b u t e d t o t h edifferences in t h e roughness and also in t h e Re number of t h e experiments. Also,
    • Effect of coexisting current 159 F=ipCDDUIUI+pCMAU KC = 5 U= U c + U m sin(mt) 2 1 "i 1 r a) KC=10 ~ 1 1 r ~i 1 1 r * b) KC = 2 0 c) 2 u c /u n 2 Uc/Un Figure 4.23 Effect of coexisting current on in-line force coefficients. Data from Sumer et al. (1992), Re = 3 X 10 4 and k/D = 4 X 1 0 " 3 . Dotted curves: Sarpkaya and Storm (1985), k/D = 1 0 - 2 and Re = 1.8 x 10 4 for KC = 10 and 3.6 x 10 4 for KC = 20. Dashed lines: Asymptotic values for steady current for k/D = 4 x 1 0 " 3 (k3/D = 10 X 10~ 3 ) and Re = 3 x 10 4 taken from Achenbach and Heinecke (1981) (see Fig. 2.11).the forces t h a t have been predicted in Sumer et al.s study are from the pressuremeasurements at the middle section of the cylinder while, in t h e study of Sarpkayaand Storm, they were measured by the force transducers over a finite length of thecylinder. Fig. 4.24 illustrates t h e influence of current on the lift coefficient. T h e liftcoefficient is defined in t h e same way as in Eq. 4.70 with Um replaced now byUc + Um- T h e figure indicates t h a t Ci, m a x decreases markedly when the current issuperimposed on the oscillatory flow. Yet, as the ratio Uc/Um increases, the liftcoefficient might be expected to approach its asymptotic value obtained for the
    • 160 Chapter J>: Forces on a cylinder in regular waves f *-"Lmax Lmax =|pCLmaxD(Um+Uc): 2 KC = 5 KC = 10 3 Uc/U„ Figure 4.24 Effect of coexisting current on lift coefficient. Data from Sumer et al. (1992). k/D = 4 x 1 ( T 3 , Re = 3 x 10 4 . Dashed lines: Asymptotic values for steady current for Re = 3 X 10 taken from Fig. 2.15 where the given r.m.s. value of the lift is multiplied by yl to obtain C i m a x current-alone case (indicated in the figure with dashed lines). Although the d a t afor KC = 5 and KC = 10 indicate t h a t this is indeed t h e case, t h e m a x i m u mvalue of t h e tested range of Uc/Um is too small to demonstrate this for KC = 20. It may be concluded from the presented results t h a t the superposition of asmall current on waves may generally reduce the force coefficients. As t h e currentcomponent of the combined waves-and-current flow becomes increased, however,the force coefficients tend to approach their asymptotic values measured for thecase of current alone.
    • Effect of angle of attack 1614.5 Effect of angle of attack It has been seen in Section 2.6 t h a t the so-called independence or cross-flow•principle (namely the normal component of force, i<V, (see Fig. 2.18) is expressedin terms of t h e n o r m a l component of t h e flow, UN, with a force coefficient which isindependent of t h e angle of attack, 8) is generally applicable for steady currents. T h e relationship expressing the independence principle, Eq. 2.14, may beextended to oscillatory flows in the form of t h e Morison equation: FN = pCDDVNUN + PCMA UN (4.75)T h e question, however, is whether the force coefficients Cu and CM a r e constants(independent of 8), in line with the steady-current case. For large KC numbers, t h e inertia portion of the force is not important.Since the oscillatory flow in this case resembles the steady current, it is there-fore expected t h a t the cross-flow principle is valid here, and hence Co may beindependent of 8. At t h e other extreme, namely for small KC numbers, on t h e other hand,the drag portion of the force is insignificant. In this case, t h e flow behaves like apotential flow, and hence the cross-flow principle must be valid here, too, meaningt h a t t h e inertia coefficient CM might be expected to approach t h e potential-flowvalue, namely CM — 2, regardless of t h e value of 8. Fig. 4.25 illustrates the effect of 8 on the force coefficients. Here KC andRe are defined in terms of the normal component of the velocity, J7;vm: RC = UN^U, a n d Re = UN^D (4 ?6) T h e d a t a apparently seem to confirm the argument put forward in thepreceding paragraphs; i.e., 1) the drag coefficient Co appears to be independentof 8 for large KC numbers (such as KC ~ 20), and 2) the inertia coefficient CMapproaches the potential-flow value, CM —* 2 for small KC numbers (such asKC ~ 8), regardless of the value of 8. T h e differences observed in the range 8 ~ KC ~ 20 in Fig. 4.25 may bea t t r i b u t e d to t h e disruption of the transverse-vortex-street regime (8 < KC < 15)for the values of angle of attack 8 = 45° a n d 8 = 60°. Even a small deviation from8 = 90° seems to influence the force coefficients. A deviation from 8 = 90° meansthat there exists a flow component parallel to t h e axis of the cylinder. This wouldeventually disrupt the transverse vortex street, leading to the observed differencesin t h e force coefficients for flow angles different from 90°.
    • 162 Chapter 4- Forces on a cylinder in regular waves 2.0 1.8 1.6 1.4 - 9 1.2 1.0 a) 0.5 i i i i i J I I I I 1 L_ 10 20 40 80 KC 4.0 3.0 2.0 1.5 1.0 b) _i L I i I . I _i_l_ 0.5 10 20 40 80 KC Figure 4.25 Effect of angle of attack on in-line force coefficients. Definitions of Co, CM, KC and Re, see Eqs. 4.75 and 4.76. The Reynolds number of the tests is such that Re/KC = 4000. (a): Sarpkaya et al. (1982). (b): Sarpkayaet al. (1982) as modified by Garrison (1985).
    • Effect of orbital motion 163 For further information about t h e effect of angle of attack, reference maybe m a d e to Chakrabarti, Tarn and Wolbert (1977), Sarpkaya, Raines and Trytten(1982) and Garrison (1985). Kozakiewicz et al. (1995) have m a d e a study of the effect of angle of attackon forces acting on a cylinder placed near a plane wall. T h e y tested three valuesof 0, namely 9 = 90°, 60° and 45°, and three values of clearance between t h ecylinder a n d t h e wall, e/D — 0, 0.1 and 1.8, e being t h e clearance for a ratherwide range of KC 4 < KC < 65. Their results indicate t h a t , for t h e tested rangeof 9, the force coefficients Co, CM and CL are practically independent of 8, evenin the range 8 ~ KC ~ 30. As noted above, t h e difference observed for thisrange of KC number for a free cylinder are due to t h e disruption of t h e transversevortex-street regime when 8 is changed from 90° to 45° and 30°. Now, in t h e caseof a near-wall cylinder, this vortex-flow regime does not exist at all, not even forthe case of perpendicular pipe (9 = 90°), owing to the close proximity of t h e wallto the pipe. Therefore, no change in the force coefficients should be expected.Sumer et al. (1991) give t h e limiting value of e/D for t h e disappearance of thetransverse-vortex-street regime for 8 = 90° as e/D = 1.7-1.8, see Section 3.4).4.6 Effect of orbital motion Until now forces on a cylinder in a plane oscillatory flow have been studied.Clearly, real waves differ from the case of plane oscillatory flow in several aspects.An i m p o r t a n t difference between t h e two cases is t h a t while t h e water particles inthe case of plane oscillatory flow travel over a straight-line trajectory, the trajec-tory of the orbital motion of water particles in the case of waves is elliptical wherethe ellipticity of t h e motion may vary between 0 (the straight-line motion) and 1(the circular motion). Hence it may be anticipated t h a t t h e forces on a cylindersubject to a real wave, may be influenced by t h e presence of the orbital motion. This section will give a detailed account of t h e subject. First, t h e vertical-cylinder case and subsequently the horizontal cylinder case will be studied. T h ecylinder diameter is assumed to be so small compared to t h e wave length t h a teffects of diffraction can b e neglected (see Chapter 6).4.6.1 Vertical cylinder Figs. 4.26 and 4.27 depict two kinds of d a t a related to t h e in-line force; onefor small Re numbers (Fig. 4.26) and t h e other for large Re numbers (Fig. 4.27),taken from Stansby, Bullock and Short (1983) and B e a r m a n et al. (1985a), re-spectively. In t h e figures, t h e plane oscillatory flow results (from Sarpkaya (1976a)
    • 16Jf Chapter 4- Forces on a cylinder in regular wavesand Justesen (1989), respectively) are also included, to facilitate comparison. T h ein-line coefficients, Co and CM, in t h e figures are defined in the same way as inEq. 4.29, U being t h e horizontal component of the velocity. o c) 10 20 KC 30 Vm Symbol v Flow Ref. Plane Sarpkaya — 0.0 oscillatoiy (1976a) flow o 0.3 Stansby a 0.5 Real etal. + 0.7 waves (1983) A 0.9 Figure 4.26 Effect of orbital motion on in-line force for vertical cylinders for small Re numbers, (a): Drag coefficient, (b): Inertia coefficient, (c): Force coefficient for the total in-line force. Sarpkaya data in (a) and (b) are for /?(= Re/KC) = 784. The Sarpkaya curve in (c) is worked out from Co and CM values given by Sarpkaya for /3(= Re/KC) = 784. In Fig. 4.26, the quantity E, denned by m E = (4.77) U„.is the parameter which characterizes the eUipticity of the orbital motion. Vm andUm are t h e m a x i m u m values of vertical and horizontal components of particle
    • Effect of orbiial motion 165 o 1Z Re = 1.5 - 5 x 1 0 " Frms • °) + 1.5r 9 + n • regular waves 6 B + random waves % 3 ***** B^n „ 0.0 i i i i i 10 15 20 25 KC Vnns v Symbol E Flow Ref. rms Justesen Plane (1989) 0.0 oscillatory Re= 5 flow 2.5 x 10 B e a r m a n et Real al. (1985a] • 0.11 - 0 . 6 5 regular Re = waves 1.5-5 x 10 5 Figure 4.27A Effect of orbital motion on in-line force for vertical cylinders for large Re numbers, (a): Drag coefficient, (b): Inertia coefficient. (c): Force coefficient for the total in-line force.velocity, respectively. In Fig. 4.27A, t h e ellipticity is given in terms of r.m.s.values of the velocity components rather t h a n the m a x i m u m values, in conformitywith the original notation of the authors (Bearman et al., 1985a). T h e quantityCj? r m s in t h e figures, on the other h a n d , is the force coefficient corresponding tothe total in-line force, defined by 1 Frms = -^pCFrmsDU, rms (4.78)in which FTms is t h e root-mean-square (r.m.s.) value of t h e in-line force per unitlength of t h e cylinder, and Ulms is t h e r.m.s. value of t h e horizontal velocity atthe level where t h e force is measured. For the small-Re-number experiments (Fig. 4.26), as far as Co and CM areconcerned, it is difficult to find any clear trend with respect to t h e ellipticity of t h e
    • 166 Chapter J: Forces on a cylinder in regular wavesmotion, t h e scatter being quite large. However, when the d a t a are plotted in terms°f Cprmsi they collapse on a narrow b a n d , with the exception of E = 0.9. Thislatter diagram indicates t h a t the total in-line force is hardly influenced by t h eellipticity of the orbital motion unless t h e ellipticity is extremely large, namelyE > 0.7 — 0.8. For such large E values t h e d a t a indicate t h a t there will be areduction in t h e total in-line force by an amount in the order of magnitude of20-30%. As for the large-Re-number experiments (Fig. 4.27A), t h e effect of orbitalmotion is indistinguishable for t h e reported range of E, namely E = 0.11 — 0.65.Also, it may be noticed t h a t t h e Cr) and CM variation obtained by Jusiesen (1989)in plane oscillatory flows (E = 0) for a Re number which lies approximately atthe centre of B e a r m a n et al.s Re range is not extremely different from t h a t ofBearman et al.s real-wave results. From the preceding discussion it may be concluded t h a t the total in-lineforce is practically uninfluenced by t h e orbital motion, unless t h e ellipticity of themotion is quite large (E > 0.7 — 0.8). In the latter case there may be a reductionin t h e total in-line force by an amount in t h e order of magnitude of 20-30%, withrespect to t h e value calculated using t h e plane oscillatory flow d a t a , meaning t h a tt h e plane-oscillatory-flow calculations remain on t h e conservative side for theseellipticity values. Fig. 4.27B presents t h e d a t a related to the lift force. Although Bearmanet al. (1985a) report t h a t the dependence on ellipticity E is indistinguishablefrom their d a t a with E ranging from 0.11 to 0.65, t h e figure indicates, however,t h a t the lift may be different from t h a t measured in the case of plane oscillatoryflow (E = 0) as measured in Justesens (1989) study. A close examination ofthe figure shows t h a t this deviation occurs in the range of KC from 7 t o 13. Asseen in Section 3.2, t h e range of KC number 7 < KC < 13, known as t h e singlepair vortex-shedding regime for plane oscillatory flows, is the range where t h e so-called transverse vortex regime prevails. T h e observed deviation from t h e planeoscillatory flow in this range of KC number may be attributed to the disruption ofthe transverse vortex street in t h e case of real waves with ellipticities different fromzero. Outside this range, however, t h e agreement between the results obtained inthe case of plane oscillatory flow and those obtained in the case of real wavesappears t o be r a t h e r good. Presumably this leads to t h e conclusion t h a t the liftforce is practically uninfluenced by t h e orbital motion with the exception of t h eKC range 7 < KC < 13, where the lift force is reduced quite considerably withrespect to t h a t experienced in t h e case of plane oscillatory flow. T h e vertical-cylinder problem has been investigated rather extensively in t h epast, Ramberg a n d Niedzwecki (1979), Chakrabarti (1980) and Sarpkaya (1984).T h e wave parameters in Chakrabartis (1980) study were such t h a t the waves werecloser to the shallow-water regime, while Ramberg and Niedzweckis were close toor in t h e deep-water regime. Nevertheless, t h e results of these two studies arein accord with Stansby et al.s study (presented in Fig. 4.26) in t h e sense t h a tthe in-line force is practically uninfluenced by t h e orbital motion in C h a k r a b a r t i s
    • Effect of orbital motion 161 R e = 1.5 - 5 x 10 V Symbol 17 mis Flow Ref. Plane Justesen 0.0 oscillatory (1989) flow Re=2.5xl0 Real Bearman D 0.11-0.65 regular et al.( 1985a) waves Re=1.5-5xl0 5 0.5 10 15 20 25 KC Figure 4.27B Effect of orbital motion on lift force (transverse force) for vertical cylinders for large Re numbers.(1980) study (small E values) while it is considerably overestimated by the plane-oscillatory-flow calculations in Ramberg and Niedzweckis study (large E valuessuch as E > 0.8 — 0.9). Sarpkaya (1984), on the other hand, simulated the orbitalmotion by oscillating t h e cylinder along its axis in a plane oscillatory flow t h a ttakes place in a direction perpendicular to the cylinder axis. Sarpkayas resultsshow a very distinct trend of t h e variation of the force coefficients Co and CM asfunction of the ellipticity parameter, E. He reports a decrease in the total forcewith increasing ellipticity.
    • 168 Chapter 4: Forces on a cylinder in regular wavesE x a m p l e 4.5: I n - l i n e force o n a v e r t i c a l pile i n t h e s u r f a c e z o n e W h e n the Morison equation is used, it will be found t h a t the in-line forceon a vertical pile is m a x i m u m at t h e level of the wave crest. However, the analysisof t h e field d a t a (Dean, Dalrymple a n d Hudspeth, 1981) show t h a t t h e force ism a x i m u m at an elevation somewhat below the water surface at t h e wave crest,becoming zero at an elevation somewhat above the wave crest (see Fig. 4.28).This observation was later confirmed by the laboratory study of T 0 r u m (1989). T h e reason behind this behaviour is t h e surface runup in front of t h e cylinderand the surface rundown at the back, presumably leading to a m a x i m u m belowthe crest elevation. T h e previously mentioned studies indicated t h a t t h e locationof the force m a x i m u m lies approximately Ull/(2g) below t h e crest level, while thelocation of zero force lies approximately U^n/(2g) above the crest level in whichUm is the m a x i m u m value of the horizontal velocity at the wave crest. As regards t h e in-line force coefficients for the region above the mean waterlevel, T 0 r u m (1989) recommends t h e following. 1) As for t h e CQ values, use Covalues as below the mean water level and 2) as for the CM values, use the CMvariation given in Fig. 4.28. U^/(2g) U£/(2g) Force, F -M ~ Values relevant to the prototype D* = J Reynolds number and Keulegan-Carpenter number U = Maximum water-particle velocity a t the crest Figure 4.28 Recommended design Co and CM values in surface zone area (T0rum, 1989).
    • Effect of orbital motion 1694.6.2 Horizontal cylinder Fig. 4.29 presents the results of B e a r m a n et al.s study (1985a) for thecase of horizontal cylinder with regard to the in-line force coefficients CD and CMfor two different Re number intervals in the post-critical Re number range. T h erange of ellipticity E in these experiments is from 0.15 to 0.75. T h e figure includesalso the results of Justesens (1989) plane-oscillatory-flow study (E = 0) for thecorresponding Reynolds numbers. Although the scatter in B e a r m a n et al.s d a t a isquite large, it is difficult to speak of any definite trend with respect to the ellipticityof the orbital motion from the data. Fig. 4.30 presents the d a t a from the samestudy (Bearman et al.s) related to the total force, namely FT = (F2 + F^)12, interms of the corresponding force coefficient defined by J V r m s = lpCTrmsDUTlms (4.79)in which FTrms=(FLs + F2Llms)1/2 (4.80)and UT.las=(uLs + VI2ms)1/2 (4.81)where F and Fi are the in-line and lift force components while U and V arethe horizontal and vertical components of the particle velocity, respectively. Thisfigure, too, shows t h a t the influence of the orbital motion on the force is notdistinguishable. For small Reynolds numbers, however, a systematic reduction inthe total in-line force with the ellipticity has been reported by Maull and Norman(1979). Maull and Normans result is reproduced in Fig. 4.31. Several investigators simulated t h e wave-induced, orbital flow around thehorizontal cylinder by driving the cylinder in an elliptical orbit in an otherwise stillwater, Holmes and Chaplin (1978), Chaplin (1981), Grass, Simons and Cavanagh(1985) and Chaplin (1988b). Chaplins (1988b) results for two different values ofthe ellipticity are plotted in Fig. 4.32. While the real-wave d a t a of Fig. 4.30 show practically no evidence aboutthe sensitivity of t h e results to orbit shape, the d a t a obtained by t h e mechanicalsimulation of the orbital flow (Fig. 4.32) indicate a systematic decrease in thetotal force with increasing wave ellipticity. This has been interpreted by Bearmanet al. (1985a) as follows. In the case of mechanical simulation of orbital flow, themotion is exactly periodic and, in the absence of any mass transport, t h e cylinderinevitably encounters its own wake, and therefore experiences a reduction in t h eincident velocity relative to the cylinder. They suggest t h a t this effect, a feature
    • 170 Chapter 4- Forces on a cylinder in regular waves o Re = 1- 3 x 1 0 Re = 3 - 5 x 10 1.5 b) 1.0 O n D 0.5 a %! [£ 0 0 1 1 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16CM 2 . 0 CM2.0 •-M 1.5 1.5 1.0 1.0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 KC KC v Symbol Flow Ref. "rms Plane Justesen 0.0 119891 oscillatory Re= 5 flow 2.5 x 10 0 0.15-0.25 Real Bearman a 0.25-0.50 regular etal. waves (1985 a) o 0.50-0.75 Figure 4.29 Effect of orbital motion on in-line force for horizontal cylinders.of the method of mechanical simulation, may be reduced by small currents or byslight irregularities in the waves. One other method of mechanical simulation of orbital flow is to oscillate thecylinder placed in a plane oscillatory flow, in a direction perpendicular to the flow.This method was used by Sarpkaya (1984). Similar to the previously mentionedwork, Sarpkaya, too, found t h a t the net result is a decrease in the total in-lineforce with increasing ellipticity.
    • Effect of orbital motion 171 ^rmst Symbol Vrms v Flow Ref. U Reg. Irreg. " rms e 0.15-0.25 Bearman Real et. al. a V 0.25 - 0.50 waves (1985 a ) o A 0.50 - 0.75 l 7 % w a .7«TT ""•* :. * 12 16 20 KC Figure 4.30 Effect of orbital motion on total (resultant) force for horizontal cylinders. The force coefficient is defined by Eqs. 4.79 and 4.81. Re = 1 — 3 X 10 5 for empty symbols and 3 — 5 X 10 5 for solid symbols. Finally, it may be mentioned t h a t , even when the force coefficients are avail-able (Fig. 4.29), the Morison equation alone provides a very poor approximationto the loading in either horizontal or vertical direction in the case of a horizontalcylinder in orbital flows (Fig. 4.33) for large KC numbers where vortex shedding
    • 172 Chapter 4- Forces on a cylinder in regular waves 12 KC Figure 4.31 Effect of orbital motion on in-line force for horizontal cylinders. The orbital motion is characterized by the ellipticity E defined by E = Vm/Um. Re = 4 x 10 3 . Maull and Norman (1979).occurs. This is because t h e vortex shedding makes a very important contributionto the loading, and obviously the Morison equation fails to represent this effect.Bearman et al. (1985a) give a detailed discussion of this aspect of t h e problem.
    • Effect of orbital motion 113 ns - ^B * FT Q X- : / > 6 E: 0.15 . 4 0.5 0.75 . ^ 2 ^ 0 i i i i i 1 1 i i i 10 14 18 KC Figure 4.32 Effect of orbital motion on total (resultant) force for horizon- tal cylinders from experiments where the orbital-motion effect is obtained by mechanical simulation, driving the cylinder in ellip- tical orbit. E = ellipticity of the orbit. Re = 1.5 - 2.2 X 10 5 . Chaplin (1988b).E x a m p l e 4.6: F o r c e s o n h o r i z o n t a l c y l i n d e r s in o r b i t a l f l o w s at low K C numbers In practice, forces on horizontal cylinders in orbital flows in the inertiaregime, particularly at r a t h e r small KC n u m b e r s , may become i m p o r t a n t . Appli-cation areas include, for example, horizontal pontoons of semi-submersibles andtension-leg platforms. In the inertia regime, the drag is insignificant, as discussedin the preceding sections. Therefore t h e total force is, to a large extent, determinedby the inertia force. T h e inertia force itself may undergo substantial reductionsin the case when the cylinder is subject to an orbital flow (or equivalently when
    • 114 Chapter ^.- Forces on a cylinder in regular waves a) Morisons equation with least squares coefficients -600 0 600 F x (N/m) b) u (m/s) Figure 4.33 (a): Horizontal cylinder: polar representation of the total force vector, averaged over about 30 waves; comparison with the least- squares Morisons equation, (b): Horizontal cylinder: polar rep- resentation of the velocity vector for the same run. Bearman et al. (1985a).it executes an orbital motion in a fluid initially at rest). This occurs at low KCnumbers; the inertia coefficient can take values as small as 50% of t h a t experi-enced in the case of planar oscillatory flow as measured by Chaplin (1984). Fig.4.34 shows t h e results of Chaplins experiments, in which real waves were used,where the diameter of the test cylinder was small compared with t h e wave length(i.e., outside the diffraction flow regime). T h e wave-induced flow was an almost
    • Effect of orbital motion 175 1 1 1 2 ^* • <^°o o •o oft o • • • o * V • 1.5 • •• * e o • • o • • oo 5fi o o o o * 8 1.0 ° CM = 2 - 0.2 KC2 > Phase lag 0.5 - - 40° i * X * X xx x * _ 20° X « x»x x «* 0 XK_— « - * • »I»IT» V 1 1 0° 3 KC Figure 4.34 Inertia coefficient for a horizontal cylinder subject to an orbital flow: •, horizontal force; o, vertical force. Phase lag of the force (ocurring at the wave frequency) with respect to the acceleration of the incident flow: X. Ellipticity, E = 0.92. /3 = 7600. L/D = 0.047 (L being the wave length). Chaplin (1984).circular orbital flow (the ellipticity, E, was 0.92). Fig. 4.34 shows t h a t CM begins to decrease already at KC about 0.5, itreaches a minium at KC about 2, and from this point onwards it increases toattain its potential-flow value, 2, at about KC = 3. It may be noticed t h a t thedrop in CM in the present case is completely different from t h a t in t h e case ofplanar flow (Figs. 4.9 and 4.10). In the latter case, for a substantial drop in CM,KC needs to be increased to such values as KC > 6 — 7. T h e observed behaviour in CM m a Y be a t t r i b u t e d to t h e steady, recirculat-
    • 176 Chapter J^: Forces on a cylinder in regular wavesing streaming which builds up around the cylinder as t h e cylinder is exposed towaves. T h e orbital flow around the cylinder may be viewed as the flow arounda cylinder which is executing an orbital motion in a fluid initially at rest. Assuch, the stirring motion of the cylinder will generate a recirculating flow in thefluid. Clearly the cylinder during its motion will encounter this flow, which is inthe same direction as the motion of the cylinder itself, meaning t h a t the inertiaforce on t h e cylinder will b e reduced. This effect is increased, as KC is increased.However, when KC reaches a critical value where the flow separates (namely,KC = 2, in the present example, see Fig. 3.15), the aforementioned recirculatingstreaming will then be disrupted by the formation of the separation vortices inthe wake, leading presumably to an increase in the CM values. W i t h the com-plete disappearence of the recirculating streaming (apparently at KC = 3), thepotential-flow value of CM (i- e -, 2) will be restored again (Fig. 4.34). A simple model to describe the inertia coefficient can be worked out on thebasis of the preceding considerations. T h e simplest case is considered; namely, thecylinder executes a circular orbital motion in a fluid initially at rest, satisfying U = Um cos(wi) and V = -Um sin(wi) (4.82)The circulation, defined as T = / v • ds, which will be generated by the stirring cmotion of the cylinder may be written as T = / y/U2 + V2 (a old) (4.83) oor, from Eq. 4.82, 2?r T = Uma I dB = 2naUm (4.84) oin which Um is the tangential velocity of the orbital motion and 2a is the strokeof the motion. Since Um = au>, then the circulation will be r = ™* (4.85)Now, the cylinder is actually subject to two kinds of flow. One is the incidentflow, i.e., the flow relative to t h e cylinder with the velocity components given inEq. 4.82. T h e other is the recirculating flow with the circulation given in Eq.4.85. T h e flow is illustrated in Fig. 4.35. First the horizontal force on the cylinderis considered. T h e flow is decomposed in the manner as sketched in Figs. 4.35band 4.35c. T h e U component of t h e flow velocity will induce an inertia force in t h ehorizontal direction, equal to 2pA U (the factor 2 being the conventional inertia
    • Effect of orbital motion 177 (a) (b) (c) Figure 4.35 Horizontal force acting on a cylinder subject to a circular orbital motion.coefficient), while the V component of the velocity combined with t h e circulationr will induce a lift force, i.e., a force perpendicular to the incident velocity V,equal to pTV, as shown in Fig. 4.35c. (The latter is known as the Magnus effect,see Batchelor 1967, p. 427). Therefore the total horizontal force, neglecting thedrag, will be F = 2pAU -pTV (4.86)or inserting F = CMPA U and Eqs. 4.82 and 4.85 into the preceding equation,CM is found CM = 2 - KC2 (4.87)or CM = 2 - 0.2 KC2 (4.88)Likewise, the inertia coefficient associated with the vertical force, namely F =CMPA V, can be worked out; it can be seen easily t h a t this will lead to the sameresult as t h a t given in the preceding equation, Eq. 4.88. T h e above equation is virtually the same equation as t h a t found by Chap-lin (1984) empirically from his force d a t a (Fig. 4.34). As seen, t h e agreementbetween this equation and the d a t a in the range 0 < KC < 2 (where the flow isunseparated) is very good. Chaplins (1984) study covered an almost idealized flow situation where thewave-induced orbital motion was more or less circular and the Reynolds numberwas small. In a later study, Chaplin (1988a) carried out similar experiments ina large scale facility where the cylinder was rather large with Re in the range
    • 178 Chapter 4: Forces on a cylinder in regular waves o *" F = ^ p C D U(U 2 + V 2 ) 2 + pC M AU Figure 4.36 The influence of elipticity, E, and the Re number on the in- ertia coefficient associated with the horizontal force for low- KC-number flows. Data: circles (Chaplin, 1984), and triangles (Chaplin, 1988a).5 x 10 4 — 4 x 10 5 and t h e waves were more realistic with ellipticity values evenbelow 0.5. T h e results of Chaplins (1988a) study are plotted in Fig. 4.36 togetherwith his earlier data. Two points may be mentioned from the figure: 1) As theellipticity increases, the reduction in the inertia force increases. 2) As the flowin the cylinder b o u n d a r y layer becomes turbulent (the large-Re number data) thereduction in CM spreads over a wider KC range (over a range of KC from 0to about 3.5 in t h e case of large-.Re-number experiments). This behaviour maybe a t t r i b u t e d t o t h e fact t h a t t h e separation is delayed by t h e turbulence in theboundary layer. In the case of elliptical orbital flows, the symmetry with respect to x andy axes will break down, therefore the vertical force will be different from thehorizontal force. Fig. 4.37 shows the results of Chaplins (1988a) large-scalefacility experiments for the inertia coefficient associated with the vertical force.The d a t a are plotted together with the corresponding d a t a of Chaplin (1984).
    • Effect of orbital motion 179 o L ^ F = -5 p C D V(U 2 + V 2 ) 2 + p C M AVCM H V V v %• I. v v v S v v • Re = 10 , E = 0.92 7 * v j v Re = (5 - 40) x 104, E s 0.5 0 2 4 6 KC Figure 4.37 The influence of ellipticity, E, and the Re number on the inertia coefficient associated with the vertical force for low KC-number flows. Data: circles (Chaplin, 1984), and triangles (Chaplin, 1988a).T h e scatter is quite extensive. However, extremely small inertia-coefficient valueshave been measured. These small values of the force are associated with the highellipticities. Chaplin (1988a) made an a t t e m p t to plot t h e d a t a in the form ofCMX versus KCy and CM*/ versus KCX, to reduce the scatter. This a t t e m p t waspartially successful. T h e issue of low AC-number orbital flows discussed in the present para-graphs has been investigated further by Chaplin (1991 and 1993b), Stansby andSmith (1991) and Stansby (1993). Chaplin (1993b) used a Navier-Stokes code,while Stansby and Smith (1991) and Stansby (1993) used the random vortexmethod, to obtain the flow field and the forces. For the latter, see Section 5.2.3.
    • 180 Chapter ^: Forces on a cylinder in regular waves4.7 Forces on a cylinder near a wall A detailed description of the oscillatory flow around a cylinder placed neara wall is given in Section 3.4. This section focuses on forces on such a cylinder,including the case of a pipeline placed in/over a scour trench.Force coefficients for a c y l i n d e r n e a r a p l a n e wall Forces on a cylinder near a plane wall and exposed to an oscillating flow hasbeen investigated quite extensively. T h e first investigation was t h a t of Sarpkaya(1976b), followed by Sarpkaya (1977a) and Sarpkaya and Rajabi (1979). Drag, in-ertia and lift coefficients on a cylinder placed at various distances from a wall weremeasured in these studies. Lundgren, Mathiesen and Gravesen (1976) measuredthe pressure distribution around a wall-mounted cylinder. Jacobsen, et al. (1984),Ali and Narayanan (1986), Justesen et al. (1987) and Sumer et al. (1991) amongothers have reported measurements regarding the effect of the wall on force coef-ficients. Forces on cylinders near a plane wall in diffraction regime are examinedin Chapter 6 and the effect of irregular waves is described in Chapter 7. Figs. 4.38 and 4.39 present t h e force-coefficient d a t a obtained in Sumeret al.s (1991) study together with Sarpkaya (1977a) and Sarpkaya and Rajabi(1979) d a t a for Re = 10 5 . Also included in the figures is Yamamoto et al.s (1974)potential-flow solution. T h e lift coefficients CLA a n d CLT are defined by &:|;K}in which FVA is the m a x i m u m value of the lift force away from the wall and Fyxthat towards the wall, e is the gap between the cylinder and the wall.Comparison with potential theory T h e experimental d a t a on CM approach t h e values predicted by t h e po-tential theory as KC — 0. Obviously, this is related to the fact t h a t , for such >small KC numbers, no separation will occur, therefore the potential-flow theorypredictions of CM must be approached, as KC goes to zero. Regarding the asymptotic behaviour of Cx a s KC — 0, for the wall- >mounted cylinder ( e / D = 0), Figs. 3.18c, 3.21d and 3.23d show t h a t the liftis always positive (directed away from the wall), in agreement with the potential-flow theory (Fig. 4.39). See also the discussion in Section 2.7 in relation to Fig.2.23. Furthermore, t h e curve representing e/D = 0 in the CLA diagram appearsto be approaching the potential-flow value, namely CLA = 4.49.
    • Forces on a cylinder near a wall 181 -D 3 n 1 1 1 1 1 1 r R e = 10 01° 1 _ _ - e/D = 0 »V .0.05 ^V- 1 1 _ l 1 I I I I I L_ 0 20 40 60 80 KC H 1 [ r -i 1 r e/D = 0 0.05 3.29 2.6 - -a—0 2.4 - o — o - 0.1 2.1 20 40 60 80 KC Figure 4.38 Drag and inertia coefficients for a near-wall cylinder. Smooth cylinder. Circles: Sumer et al. (1991) (Re = 0.8 - 1.1 X 10 5 ); A: e/D = 1; V: e/D = 0.1, Sarpkaya (1977a) (Re = 10 5 ); - - - - -: e/D = 0, Sarpkaya and Rajabi (1979) (Re = l - 1.1 X 10 5 ). The asymptotic values of CM for KC — 0 indicated in the figure * are the potential-flow solutions due to Yamamoto et al. (1974), reproduced here from Fig. 4.4 where Cm = CM — 1. However, for a cylinder placed near the wall, even with an extremely smallgap ratio such as e/D ~ 0.05, the lift alternates between successive positive andnegative peaks (Figs. 3.18b, 3.21c, 3.23c and 3.24c,d). T h e positive peak in the liftis associated with the movement of the lee-wake vortex over the cylinder during the
    • 182 Chapter J^: Forces on a cylinder in regular wavesflow reversals, while the negative peak in the lift is associated with the formationof lee-wake vortex and the high-speed flow in the gap between t h e cylinder andthe wall as discussed in Section 3.4). (e/D = 0) a) b) ~1 1 1 1 1 1 1 1— 0 20 40 60 80 KC Figure 4.39 Lift-force coefficient for a near-wall cylinder. Symbols are the same as in the previous figure. The asymptotic values of Ci for KC — 0 indicated in the figure are the potential-flow solutions » due to Yamamoto et al. (1974). From t h e discussion in Section 2.7 in relation to Fig. 2.23, it is apparentt h a t the potential-flow theory in the case of near-wall cylinder does not predicta positive lift but rather a negative lift. T h e values calculated from the poten-tial flow theory for t h e gap ratios e/D — 0.05,0.1 and 1 are indicated in Fig.4.39b. Apparently, as KC —• 0, the experimental results seem to be approachingthe potential-flow values for e/D = 0.05 and 0.1. However, for e/D = 1, theexperimental CLT values are much lower t h a n the potential-flow value, namelyCLT S - 0 . 1 .
    • Forces on a cylinder near a wall 18SInfluence of gap ratio From Figs. 4.38 and 4.39, t h e d a t a indicate t h a t t h e force coefficients CD,CM and CLA increase as the gap ratio decreases. This is also true for CLT forsmall KC(0(W)). For large KC, however, no clear trend appears. These resultsgenerally agree with those of other investigators such as Sarpkaya (1976b, 1977a),Ali and Narayanan (1986) and Justesen et al. (1987). OI» e 3- e / D = 0.05 R o u g h ( k s / D = 10 ) Smooth 2- - R o u g h ( k s / D = 10 ) Smooth 0 20 40 60 ioT KC Figure 4.40 Influence of roughness on drag coefficient. Re = 0.8 X 10 5 1.1 X 10 s . Sumer et al. (1991).Influence of roughness Figs. 4.40-4.42 compare the force coefficients obtained for t h e smooth andrough cylinders of Sumer et al.s (1991) study for the gap ratios e/D = 1 and 0.05.Figure 4.40 indicates t h a t CD increases substantially when t h e cylinder surfacechanges from smooth to rough. This is consistent with Sarpkayas (1976b) wall-free cylinder d a t a corresponding t o Re = 10 5 . Figure 4.41 indicates t h a t CMdoes not change significantly with the change of surface roughness for e/D = 1.
    • 184 Chapter 4: Forces on a cylinder in regular wavesHowever, for e/D = 0.05, the inertia coefficient increases markedly when thesurface of the cylinder changes from smooth to rough. This may be a t t r i b u t e d tot h e retarding effect of t h e b o u n d a r y layer at t h e wall side of t h e cylinder whichmay become significant for t h e inertia coefficient for small gap ratios such as 0.05. Fig. 4.42 shows t h a t no significant change occurs in t h e lift coefficients whenthe surface is changed from smooth to rough. This result appears to be consistentwith Sarpkayas (1976a) wall-free cylinder results and also with Sarpkaya andRajabis (1979) wall-mounted cylinder results. OI° 77777777X7 e / D = 0.05 Rough (ks/D =102) 4- Smooth 2.6 — 2 e/D = 1 2.1 — •^=^~~_-^- R o u g h ( k s / D = 10 ) Smooth 0 20 40 60 80 KC Figure 4.41 Influence of roughness on inertia coefficient. Re = 0.8 X 10 — 1.1 X 10 . Asymptotic values for KC —» 0 are Yamamoto et al.s (1974) potential flow solutions. Sumer et al. (1991).Influence of Re This was studied by Yamamoto and Nath (1976), and Sarpkaya (1977a).Both studies indicate t h a t the way in which the force coefficients change with Reis much the same as in the case of wall-free cylinder (Figs. 4.11 and 4.18).
    • Forces on a cylinder near a wall 185 e / D = 0.05 Rough (ks/D = 10 Smooth e/D = 1 2- 20 40 60 80 KC 0 20 40 60 80 LT 0 Figure 4.42 Influence of roughness on lift-force coefficients for smooth and rough near-wall cylinders. Re = 0.8 X 10 5 — 1.1 X 10 5 . Sumer et al. (1991).Wall-mounted cylinder (e/D = 0) Although the force coefficients for a wall-mounted cylinder are given earlierin Figs. 4.38 and 4.39, the covered KC range was somewhat limited. Fig. 4.43 gives the force coefficients, covering a much broader range ofKC number, up to 170 (Bryndum, Jacobsen a n d Tsahalis, 1992). T h e figure also
    • 186 Chapter 4- Forces on a cylinder in regular waves y,/7///. l . l . l . i . I i I i 1 1 1 1 1 1 1 1 1 1 f """ • " ~ ^ 3.29 Pot.-flow value . 1 , 1 . 1 , 1 , 1 . 1 , 1 . 1 4.49 Pot.-flow value 0 20 100 KC Figure 4.43 Force coefficients for a wall-mounted cylinder. Re = (0.5 - 3.6) x 10 5 . Bryndum et al. (1992).
    • Forces resulting from breaking-wave impact 187illustrates the surface-roughness influence. Bryndum et al. examined also otheraspects of the problem such as the influence of co-existing current, the Fouriercoefficients and phases for the drag and the lift forces and the "extreme" forcecoefficients, defined by CH^=[FH{tj^J(pDUl) v (4.90)for the horizontal and vertical force components, respectively. An extensive com-parison of data was made by Bryndum et al., covering the laboratory tests reportedby Sarpkaya and Rajabi (1979), the laboratory tests carried out at the NorwegianHydrodynamic Laboratories (NHL) and reported in NHL (1985) and the fieldexperiments undertaken off the coast of Hawaii and reported by Grace and Zee(1979).Force coefficients for pipelines. Partially buried pipes and pipes intrenches Fig. 4.44 depicts the force coefficients corresponding to the case of a par-tially buried pipeline, while Fig. 4.45 illustrates the influence of a trench hole(Jacobsen, Bryndum and Bonde, 1989). The force coefficients Coo, CMO and CLOin the figures are those for a pipe resting on a plane bed (Fig. 4.43). As seen,the force coefficients are generally reduced, in some cases quite substantantially.The reduction in the force coefficients is due to sheltering effect, as discussed inSection 2.7 in relation to forces on pipelines in trench holes in the case of steadycurrent (Fig. 2.32). The larger the sheltering effect, the larger the reduction inthe force coefficients. Jacobsen et al. investigated also the influence of co-existingcurrent on the force coefficients for the partially-buried-pipe case, which indicatedthe same kind of trend as in Fig. 4.44. In addition to the aforementioned cases,Jacobsen et al. carried out tests on pipelines sliding on the bed.4.8 Forces resulting from breaking-wave impact The impact forces on marine structures such as breakwaters, sea walls, piles,etc. generated by breaking waves can attain very large values. Works by Kjeldsen,T0rum and Dean (1986) and Basco and Niedzwecki (1989) show, for instance, thatplunging wave forces on a pile can be a factor of 2-3 times larger than the ordinaryforces with waves of comparable amplitudes. Before considering the vertical-pile case, we shall study a simpler case,namely the case of a vertical wall exposed to the action of the impact of a plunging
    • 188 Chapter 4: Forces on a cylinder in regular waves 1 1 1 1 1 1 1 o o • o < o w - o oo • o < : o • o < - CD - 0 «0 < - O ~ •O1 . o < < o o . A < CN r° "11, 1 1 1 1 1 1 i i i i 1 1 1 < o o • o lid . o - o - 00 • o <J - • 0 < o - CD • 0 < O . o - s •o • o < • o < o " 2 < CN O • o 1 1 1 ," f , I > I 100 Q ~ d d « - o 00 < o CO < O o Q o o o • c » o < CN o _L_L J_l_ in d
    • Forces resulting from breaking-wave impact 189 1 1 1 1 | 1 1 1 " mtfieo o- H/D inqq - - o»-4 - _ 0 • -4 O * -4 - - - O • -4 - 0 • -4 " _ 0 -J - O • 4 - o -5 0 *4 O »-4 * O IT" 1 1 i i - i i-l I T — I — | — r— i — i — i — u- Hi - - 04 • - - 0-4 • _ O -4» - - - 0 » -4 - - o • -4 •O 4 _ u 0» -4 - _s cm •* 3" _ o 1 1 i i i i T i T i r 1 1 o- « - - 0» -4 - _ O* -4 _ - O* -4 . - O • -4 - o O 4-4 - - D O »-4 - O • -4 - Q O • -4 .o I • 1 1 10 6
    • 190 Chapter 4- Forces on a cylinder in regular waves / / Figure 4.46 Breaking-wave profiles until the instant of impact. Vertical- wall case. Chan and Melville (1988).breaker, sketched in Fig. 4.46. T h e figure illustrates t h e breaking-wave profiles atprogressive times with interval A t where A t is in the order of magnitude of 0.02T,T being the wave period, C h a n and Melville (1988). As the wave approachesthe wall, the breaking wave (the wave profile corresponding to time t + 3At) willimpinge on the wall at a certain location, Location M. T h e impingement of thewater on the wall will exert an impulsive pressure on the wall at M, the impactpressure. As t h e wave progresses, the impact pressure will be experienced on thewall over a larger and larger wall area. Fig. 4.47 shows t h e time series of the pressure measured at the point ofinitial impact. As seen, the pressure increases impulsively, and then it exhibits anoscillatory character as it decreases after t h e peak. While the impulsive increase isdue to the impact, called hammer shock (Lundgren, 1969), t h e oscillatory characterof the pressure variation is linked with t h e air t r a p p e d in the water during thecourse of impact of t h e water mass (see, for example Lundgren, 1969, Chan andMelville, 1988). First of all, t h e impact characteristics are dependent on the particular lo-
    • Forces resulting from breaking-wave impact 191 i{ p/(pc2) 10- 5- 1 0 J V^____ 1 1 1 _ * 0.01 T Figure 4.47 Pressure time series at the point of initial impact. Vertical wall. Chan and Melville (1988).cation of t h e wall relative t o t h e location of t h e wave breaking. Fig. 4.48 summa-rizes the impact characteristics with t h e wall location. T h e most critical locationis where the wave plunging develops just before t h e impact (Fig. 4.48c). Chanand Melville reports t h a t , in this case, the direction of the crest is approximatelyhorizontal. No impact pressures are generated for the locations in Figs. 4.48a and4.48f. This is simply because wave breaking occurs too late for t h e case depictedin Fig. 4.48a, a n d it occurs too early for that in Fig. 4.48f. Second, pressures at the critical location are the highest. T h e normalizedm a x i m u m impact pressures, p/(pc2), typically range from 3 to 10 in which c is thewave celerity, c = L/T, with the corresponding rise time being in t h e range 0.0005Tto 0.002T. T h e obtained peak pressures are comparable to those of the others (seeTable 4.1). T h e broad range of the measured peak pressure, a feature common toall the other studies as well (see Table 4.1), indicates the strong randomness in theprocess. This is due partly to the randomness in the wave breaking process (andhence due t o the randomness in the dynamics of the t r a p p e d air) a n d partly to therandomness in the air-entrapment process. This will result in strong "turbulence"in the measured pressure signal, revealing the observed broad range of pressures. Third Fig. 4.49 displays the impact pressure distribution over t h e depthat the location where the largest pressure peaks are experienced. Here, z = 0
    • 192 Chapter 1^: Forces on a cylinder in regular waves Figure 4.48 Schematics of breaking waves incident on a vertical wall. Chan and Melville (1988).is the stationary water level. As seen, the m a x i m u m pressure occurs at aboutz/L = 0.05. Fourth, Chan and Melvilles results as well as t h e results of the others (Table4.1) indicate t h a t t h e impact pressure scales with pc2. This can be inferred fromsimple impulse-momentum considerations. T h e impulse-momentum equation forthe control volume shown in Fig. 4.50 can, to a first approximation, be written as pqc = pa (4.91)in which q is the rate of flow per unit width, q = cA, and a and A are thecorresponding areas. Hence, a crude estimate of the pressure can be obtainedfrom the preceding equation as p/(pc2) = A/a, illustrating t h a t the pressure scaleswith pc2. Clearly, the ratio A/a is much larger t h a n unity, since at the instant ofimpingement, the impact occurs through the focusing of the incident wave frontonto the wall (Chan and Melville, 1988, p.127), revealing the range observed inthe experiments (Table 4.1), namely p/(pc2) = 3 — 10. Chan, Cheong and Tan (1995) extended Chan and Melvilles study to thecase of v e r t i c a l c y l i n d e r s . Figs. 4.51-4.53 display three sequences of pho-tographs, illustrating the way in which the incoming wave impinges on the cylinder.In Fig. 4.51, the wave impinges on the cylinder before wave breaking occurs, while,in Fig. 4.53 it impinges on the cylinder long after wave breaking occurs. Therefore,
    • Forces resulting from breaking-wave impact 193 Table 4.1 Comparison of peak impact pressures. Typical range of peak pressures Investigator Pm/{PC2) StructureKjeldsen & Myrhaug 1-2 Vertical plate suspended above SWL(1979) (deep water).Kjeldsen (1981) 1-3 Inclined plate suspended above SWL (deep water).Ochi & Tsai (1984) 1.4 Surface-piercing cylinder (deep water).Bagnold (1939) 11-40 Surface-piercing plate on a sloping beach. (highest 90)Hayashi h Hattori 3-15 Surface-piercing plate on a sloping beach.(1958)Weggel b. Maxwell 8-20 Surface-piercing plate on a sloping beach.(1970) (highest 40)Kirkgoz (1982) 8-20 Surface-piercing plate on a sloping beach.Blackmore & Hewson 0.5-4 Seawall (prototype structure).(1984)Chan & Melville 3-10 Surface-piercing plate (deep water).(1988) (highest 21)in these two cases, no significant impact pressure develops, as demonstrated by thepressure measurements of C h a n et al. (1995), whereas, in Fig. 4.52 (the criticalcase, somewhat similar to t h a t given in Fig. 4.48c), the impingement of breakingwave is such t h a t very high impulsive impact pressures are generated. T h e pres-sure measurements of Chan et al. (1995) indicate that the impact pressure is thehighest at the instant corresponding to Fig. 4.52b. Fig. 4.54 gives t h e measured time series of pressure on the u p s t r e a m edge ofthe cylinder. T h e pressure characteristics are basically similar to those observedfor the vertical wall situation. Chan et al. (1995) observed t h a t the impact pressure decreased graduallywith the azimuthal angle, #, where 6 = 0 corresponds to the upstream edge ofthe cylinder. Also observed is the fact t h a t the occurrence of peak pressures isdelayed for locations of larger azimuthal angles, consistent with the motion of thewave crest around the cylinder. T h e observed extent of the area where the impactpressures p/(pc2) are larger t h a n 3 is —22.5° < 8 < +22.5°. One final point as
    • 194 Chapter 4- Forces on a cylinder in regular waves 1 1 0.08 - " 0.06 - - * H 0.04 Q S 0.02 - 1 1 10 15 p/(pc2) Figure 4.49 Vertical distribution of impact pressure at the location where the largest pressure peaks are experienced. Vertical wall. Chan and Melville (1988). Control v o l u m e Figure 4.50 Definition sketch for the application of the impulse-momentum principle.
    • Forces resulting from breaking-wave impact 195regards the azimuthal variation of the impact pressures is t h a t it is not always0 = 0° where m a x i m u m impact pressures occur; C h a n et al. (1995) observed t h a tthe maximum pressures can occur off t h e symmetry line 8 = 0°, at such 8 valuesas high as 15°. This is due to t h e turbulence referred to earlier. T h e resulting impact force was estimated in C h a n et al.s (1995) study by / = / J pr0 cos(8)d8dz (4.92) JAz JBin which r0 is the radius of the cylinder, and Az is t h e vertical extent of the impactzone (cf. Fig. 4.49). Subsequently, t h e force coefficients Cs are calculated from: / = ^PCS Az Dc2 (4.93)For example, t h e Cs value obtained at the instant of peak pressure occurrence at0 = 0° is Ca = 7.0, while t h a t obtained a t t h e instant of peak pressure occurrenceat 8= 15° is Cs = 11.4. Fig. 4.54 landscape figure caption in test-hj An estimate of the force coefficient Cs can be m a d e , adopting t h e method ofKaplan and Silbert (1976). T h e in-line impact force per unit height of the cylinderin the impact zone (Fig. 4.55) just after the impact will be F=^fl (4,4)in which t h e drag force and t h e Froude-Krylov force are neglected, since we areinterested in t h e force at t h e instant of impact (t, x —• 0). Here, U is the horizontalcomponent of the velocity and m is the hydrodynamic mass, corresponding tothe hatched area (section a-a) in Fig. 4.55. T h e right hand-side of the precedingequation can be written as _ ,dU rT/dmdxSince the velocity U can be considered constant, equal to t h e wave celerity, c, theequation becomesm is given by Taylor (1930) (see Kaplan and Silbert, 1976) 3 / ! 2 2TI- (1-COS0) TT., „, , „ „N (4.97) ~J-(2^8y+3{1-cos^ + ^ e - e At the instant of impact (x —> 0), it can be shown t h a t dm ! -pr0n (4.98)
    • Chapter 4- Forces on a cylinder in regular wavesFigure 4.51 Development of wave plunging when cylinder is located at x/L = 4.764, At = (a) 0, (b) 0.01 s (0.008T), (c) 0.02 s (0.016T), (d) 0.03 s (0.023T). X is the distance from the wave pedal. Chan et al. (1995) with permission - see Credits.
    • Forces resulting from breaking-wave impactFigure 4.52 Development of wave plunging when cylinder is located at x/L - 4.885, A t = (a) 0, (b) 0.01 s (0.008T), (c) 0.02 s (0.016T), (d) 0.03 s (0.023T). Chan et al. (1995) with permission - see Credits.
    • Chapter 4- Forces on a cylinder in regular wavesFigure 4.53 Development of wave plunging when cylinder is located at x/L = 5.047, A t = (a) 0, (b) 0.01 s (0.008T), (c) 0.02 s (0.016T), (d) 0.03 s (0.023T). Chan et al. (1995) with permission - see Credits.
    • a) b) 0 18 12pc"1 6 0 JL^.. . A«_ 4- • libOMM ^ ?L. ^ - A— J^- -JV-/!^W— B^£»0»J ^ a -» «3"lrt»-..i J L J L _L 0 t/T 0.032 Figure 4.54 Simultaneous pressure time histories recorded from repeated experiments (a, x/L = 4.885. Tjjn is the undisturbed crest elevation. Chan et al. (1995).
    • 200 Chapter li: Forces on a cylinder in regular wavesUsing the force coefficient definition in Eq. 4.93 and recalling t h a t the force Fis calculated per unit height of the impact zone, the force coefficient is obtainedas C3 = 7r, the commonly used value in the empirical models (Goda et al., 1966(referred to in the paper by Sawaragi and Nochino, 1984) and Wiegel, 1982). Asseen, the experimentally obtained values of the force coefficient Cs is a factor of2-4 larger t h a n the theoretical estimate of Cs. (Similar results were obtained alsoby Sawaragi and Nochino, 1984). This may be a t t r i b u t e d partly to the effect oftrapped air. Section a-a Figure 4.55 Definition sketch. T h e previously mentioned studies have been extended by Chan, Cheong andGin (1991) to the case of a horizontal beam, and by C h a n (1993) to the case of alarge horizontal cylinder in the splash zone where the structures were placed justabove t h e still water level and exposed to plunging waves. Oumeraci, Klammerand Partenscky (1993) have, for the case of a vertical wall simulating a caissonbreakwater, demonstrated t h a t the impact pressure changes, depending on thebreaker type. Criteria have been developed for wave breaking and breaker-typeclassification in this latter study. T h e breaking-wave impact pressure has beenfurther elaborated by researchers such as Hattori, Arami and Yui (1994), Chan
    • References 201(1994), Goda (1994) and Oumeraci and Kortenhaus (1994) in conjunction withthe vertical-wall breakwaters. Sawaragi and Nochino (1984) studied the case of avertical cylinder for b o t h t h e spilling type breaker and the plunging type breaker;the former gave smaller peak pressures in most cases. Tanimoto, Takashi, Kanekoand Shiota (1986) studied the impact forces of breaking waves on an inclined pile.Endresen and T 0 r u m (1992) and Yuksel and Narayanan (1994) studied breaking-wave forces on pipelines on the seabed.REFERENCESAchenbach, E. and Heinecke, E. (1981): On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 x 10 3 to 5 x 10 6 . J. Fluid Mech., 109:239-251.Ali, N. and Narayanan, R. (1986): Forces on cylinders oscillating near a plane boundary. Proc. 5th Int. Offshore Mechanics &: Arctic Engineering (OMAE) Symp., Tokyo, J a p a n , 111:613-619.Anatvirk, A. (1991): An experimental investigation to measure hydrodynamic forces at small amplitudes and high frequencies. Applied Ocean Research, 13(4):200-208.Bagnold, R.A. (1939): Interim report on wave pressure research. J. Inst. Civil Engrs., 12:201-226.Basco, D.R. and Niedzwecki, J.M. (1989): Breaking wave force distributions and design criteria for slender piles. O T C 6009, pp. 425-431.Batchelor, G.K. (1967): An Introduction to Fluid Dynamics. Cambridge Univer- sity Press.Bearman, P.W., Chaplin, J.R., Graham, J.M.R., Kostense, J.K., Hall, P.F. and Klopman, G.(1985a): T h e loading on a cylinder in post-critical flow be- neath periodic and r a n d o m waves. Proc. 4th Int. Conf., In: Behaviour of Offshore Structures, Delft, Elsevier, Ed. J.A. Battjes, Developments in marine technology, 2, p p . 213-225.Bearman, P.W., Downie, M.J., Graham, J.M.R. and Obasaju, E.D. (1985b): Forces on cylinders in viscous oscillatory flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 154:337-356.
    • 202 Chapter 4- Forces on a cylinder in regular wavesBearman, P.W. and Obasaju, E.D. (1989): Transverse forces on a circular cylinder oscillating in-line with a steady current. Proc. 8th Int. Conf. on Offshore Mechanics and Arctic Engineering, O M A E , T h e Hague, March 19-23, 1989, 2:253-258.Blackmore, P.A. and Hewson, P.J. (1984): Experiments on full scale wave impact pressures, Coastal Engrg., 8:331-346.Bryndum, M.B., Jacobsen, V. and Tsahalis, D.T. (1992): Hydrodynamic forces on pipelines: Model tests. J. Offshore Mechanics and Arctic Engineering, Trans. ASME, 114:231-241.Chakrabarti, S.K., Tarn, W.A. and Wolbert, A.L. (1977): Wave forces on inclined tubes. Coastal Engineering, 1:149-165.Chakrabarti, S.K. (1980): In-line forces on a fixed vertical cylinder in waves. J. Waterway, Port, Coastal and Ocean Div., ASCE, 106(WW2):145-155.Chan, E.S. (1993): Extreme wave action on large horizontal cylinders located above still water level. Proc. 3rd Int. Offshore and Polar Eng. Conf., Singapore, 6-11 J u n e , 1993, 111:121-128.Chan, E.S. (1994): Mechanics of deep water plunging-wave impact o vertical struc- tures. Coastal Engineering, 22(1,2):115-134.Chan, E.S. and Melville, W.K. (1988): Deep water plunging wave pressures on a vertical plane wall. Proc. R. S o c , London, A417:95-131.Chan, E.S., Cheong, H.F. and Gin, K.Y.H. (1991): Wave impact loads on horizon- tal structures in the splash zone. Proc. I S O P E 91, Edinburgh, 3:203-209.Chan, E.S., Cheong, H.F. and Tan, B.C. (1995): Laboratory study of plunging wave impacts on vertical cylinders. Coastal Engineering, 25:87-107.Chaplin, J.R. (1981): Boundary layer separation from a cylinder in waves. Proc. International Symposium on Hydrodyn. in Ocean Engrg., Trondheim, 1981, 1:645-666.Chaplin, J.R. (1984): Non-linear forces on a horizontal cylinder b e n e a t h waves. J. Fluid Mech., 147:449-464.Chaplin, J.R. (1988a): Non-linear forces on horizontal cylinders in the inertia regime in waves at high Reynolds numbers. Proc. Int. Conf. on Behaviour of Offshore Structures (BOSS 88), Trondheim, J u n e 1988, 2:505-518.
    • References 203Chaplin, J.R. (1988b): Loading on a cylinder in uniform oscillatory flow: P a r t II - Elliptical orbital flow. Applied Ocean Research, 10(4): 199-206.Chaplin, J.R. (1991): Loading on a horizontal cylinder in irregular waves at large scale. Int. J. of Offshore and Polar Engrg., Dec. 1991, l(4):247-254.Chaplin, J.R. (1993a): P l a n a r oscillatory flow forces at high Reynolds numbers. J. Offshore Mech. and Arctic Eng., ASME, 115:31-39.Chaplin, J.R. (1993b): Orbital flow around a circular cylinder. P a r t 2. Attached flow at larger amplitudes. J. Fluid Mech., 246:397-418.Dean, R.G., Dalrymple, R.A. and Hudspeth, R . T . (1981): Force coefficients from wave projects I and II. D a t a including free-surface effects. Society of Petroleum Engineers Journal. December 1981, p p . 777-786.Endresen, H.K. and T 0 r u m , A. (1992): Wave forces on a pipeline through the surf zone. Coastal Engineering, 18:267-281.Garrison, C.J. (1985): Comments on the cross-flow principle and Morisons equa- tion. J. Waterway, Port, Coastal and Ocean Eng., ASCE, 111(6):1075-1079.Goda, Y., Haranaka, S. and K i t a h a t a , M. (1966): Study on impulsive breaking wave forces on piles. Rep. Port Harbour Res. Inst., 6(5):l-30.Goda, Y. (1994): Dynamic response of upright breakwaters to impulsive breaking wave forces. Coastal Engineering, 22(1,2):134-158.Grace, R.A. a n d Zee, G.T.Y. (1981): Wave forces on rigid pipes using ocean test data. J. Waterway, Port, Coastal and Ocean Division, ASCE, 107(WW2):71-92.Grass, A.J., Simons, R.R. and Cavanagh, N.J. (1985): Fluid loading on horizontal cylinders in wave type orbital oscillatory flow. Proc. 4th Offshore Mechanics and Arctic Engrg. Symp., Dallas, TX., 1:576-583.Hansen, E.A. (1990): Added mass and inertia coefficients of groups of cylinders and of a cylinder placed near an arbitrarily shaped seabed. Proc. 9th Offshore Mechanics and Arctic Engrg., Houston, T X , Vol. 1, P a r t A, p p . 107-113.Hattori, M., Arami, A. and Yui, T. (1994): Wave impact pressure on vertical walls under breaking waves of various types. Coastal Engineering, 22(l,2):57-78.
    • 204 Chapter Jf-. Forces on a cylinder in regular wavesHayashi, T. and Hattori, M. (1958): Pressure of t h e breaker against a vertical wall. Coastal Engineering in J a p a n , 1:25-37.Holmes, P. and Chaplin, J.R. (1978): Wave loads on horizontal cylinders. Proc. 16th International Conf. on Coastal Engrg., Hamburg, 1978, 3:2449-2460.Jacobsen, V., Bryndum, M.B. and Freds0e, J. (1984): Determination of flow kine- matics close to marine pipelines and their use in stability calculations. Proc. 16th Annual Offshore Technology Conf., Paper O T C 4833, 3:481-492.Jacobsen, V., B r y n d u m , M.B. and Bonde, C. (1989): Fluid loads on pipelines: Sheltered or sliding. Proc. 21st Annual Offshore Technology Conf., Paper O T C 6056, 3:133-146.Jacobsen, V. and Hansen, E.A. (1990): T h e concepts of added mass and inertia forces and their use in structural dynamics. Proc. 22nd Annual Offshore Technology Conf., Houston, TX, May 7-10, 1990, Paper O T C 6314, 2:419- 430.Jonsson, I.G. (1990): Wave Current Interactions. In: T h e Sea, eds. B. Le Mehaute and D.M. Hanes, Wiley-Interscience, N.Y., Chapter 9A:65-120.Justesen, P., Hansen, E.A., Freds0e, J., B r y n d u m , M.B. and Jacobsen, V. (1987): Forces on and flow around near-bed pipelines in waves and current. Proc. 6th Int. Offshore Mechanics and Arctic Engrg. Symp., ASME, Houston, T X , March 1-6, 1987, 2:131-138.Justesen, P. (1989): Hydrodynamic forces on large cylinders in oscillatory flow. J. Waterway, Port, Coastal and Ocean Engineering, ASCE, 115(4):497-514.Justesen, P. (1991): A numerical study of oscillating flow around a circular cylin- der. J. Fluid Mech., 222:157-196.Kaplan, P. and Silbert, M.N. (1976): Impact forces on platform horizontal mem- bers in the splash zone. 8th Annual Offshore Technology Conf., Houston, TX, May 3-6, 1976, O T C 2498, p p . 749-758.Kasahara, Y., Koterayama, W. and Shimazaki, K. (1987): Wave forces acting on rough circular cylinders at high Reynolds numbers. Proc. 19th Offshore Technology Conf., Houston, T X , O T C 5372, 1:153-160.Keulegan, G.H. and Carpenter, L.G. (1958): Forces on cylinders and plates in an oscillating fluid. J. Research of the National Bureau of Standards, Vol. 60, No. 5, Research paper 2857, p p . 423-440.
    • References 205Kirkgoz, M.S. (1982): Shock pressure of breaking waves on vertical walls. J. Waterway, Port, Coastal and Ocean Div., ASCE, 108(WWl):81-95.Kjeldsen, S.P. and Myrhaug, D. (1979): Breaking waves in deep water and resul- tant wave forces. Proc. 11th Offshore Tech. Conf., Houston, T X , paper 3646, p p . 2515-2522.Kjeldsen, S.P. (1981): Shock pressures from deep water breaking waves. Proc. Int. Symp. on Hydrodynamics, Trondheim, Norway, p p . 567-584.Kjeldsen, S.P., T0rum, A. and Dean, R.G. (1986): Wave forces on vertical piles caused by 2 and 3 dimensional breaking waves. Proc. 20th Int. Conf. Coastal Engineering, Taipei, ASCE, New York, p p . 1929-1942.Kozakiewicz, A., Freds0e, J. and Sumer, B.M. (1995): Forces on pipelines in oblique attack. Steady current and waves. Proc. 5th Int. Offshore and Polar Engineering Conf., T h e Hague, Netherlands, J u n e 11-16, 1995, Vol. 11:174-183.Lundgren, H. (1969): Wave shock forces: An analysis of deformations and forces in t h e wave and in the foundation. Research and Wave Action. Proc. Symposium.Delft, Vol. 2, Paper 4.Lundgren, H., Mathiesen, B. and Gravesen, H. (1976): Wave loads on pipelines on the seafloor. Proc. 1st Int. Conf. on the Behaviour of Offshore Structures, BOSS 76, 1:236-247.Maull, D.J. and Milliner, M.C. (1978): Sinusoidal flow past a circular cylinder. Coastal Engineering, 2:149-168.Maull, D.J. and Norman, S.G. (1979): A horizontal circular cylinder under waves. Proc. Symp. on Mechanics of Wave-Induced Forces on Cylinders, Bristol, ed. T.L. Shaw, P i t m a n , pp. 359-378.Milne-Thomson, L.M. (1962): Theoretical Hydrodynamics. Macmillan.Moe, G. and Verley, R.L.P. (1980): Hydrodynamic damping of offshore structures in waves and current. 12th Annual Offshore Technology Conf., Paper No. O T C 3798, Houston, T X , May 5-8, 1980, 3:37-44.Morison, J.R., OBrien, M.P., Johnson, J . W . and Schaaf, S.A. (1950): T h e forces exerted by surface waves on piles. J. Petrol. Technol., Petroleum Transac- tions, AIME, (American Inst. Mining Engrs.), 189:149-154.
    • 206 Chapter ^: Forces on a cylinder in regular wavesNHL (Norwegian Hydrodynamic Laboratories) (1985): Design of Pipelines to Re- sist Ocean Forces. Final Report on Joint Industry R & D Program, 1985.Ochi, M.K. a n d Tsai, C.H. (1984): Prediction of impact pressure induced by breaking waves on vertical cylinders in r a n d o m seas. Appl. Ocean Res., 6:157-165.Oumeraci, H., Klammer, P. a n d Partenscky, H.W. (1993): Classification of break- ing wave loads on vertical structures. ASCE, J. Waterway, Port, Coastal and Ocean Engineering, 119(4):381-396.Oumeraci, H. and Kortenhaus, A. (1994): Analysis of the dynamic response of caisson breakwaters. Coastal Engineering, 22(1,2):159-182.Peregrine, D.H. (1976): Interaction of water waves and currents. Advances in Applied Mechanics, 16:9-117.Ramberg, S.E. and Niedzwecki, J.M. (1979): Some uncertainties a n d errors in wave force computations. Proc. 11th Offshore Technology Conf., Houston, T X , 3:2091-2101.Rodenbusch, G. a n d Gutierrez, C.A. (1983): Forces on cylinders in two- dimensional flow. Tech. Report, Vol. 1, BRC 13-83, Bellaire Research Center (Shell Development Co.), Houston, T X .Sarpkaya, T. (1976a): In-line and transverse forces on smooth a n d sand-roughened cylinders in oscillatory flow at high Reynolds numbers. Naval Postgraduate School, Monterey, CA, Tech. Rep. NPS-69SL76062.Sarpkaya, T. (1976b): Forces on cylinders near a plane b o u n d a r y in a sinusoidally oscillating fluid. Trans. ASME, J. Fluids Engng., 98:499-505.Sarpkaya, T. (1977a): In-line and transverse forces on cylinders near a wall in oscillatory flow at high Reynolds numbers. Proc. 9th Annual Offshore Technology Conf., Houston, T X , Paper O T C 2898, 3:161-166.Sarpkaya, T. (1977b): In-line a n d transverse forces on cylinders in oscillatory flow at high Reynolds numbers. Jour. Ship Research, 21(4):200-216.Sarpkaya, T. and Rajabi, F . (1979): Hydrodynamic drag on bottom-mounted smooth and rough cylinders in periodic flow. Proc. 11th Annual Offshore Technology Conf., Houston, T X , Paper O T C 3761, 2:219-226.
    • References 207Sarpkaya, T. (1981): Morisons Equation a n d t h e Wave Forces on offshore struc- tures. Naval Civil Engineering Laboratory Report, CR82.008, Port Huen- eme, CA.Sarpkaya, T. a n d Isaacson, M. (1981): Mechanics of Wave Forces on Offshore Structures. Van Nostrand Reinhold Company.Sarpkaya, T., Raines, T . S . a n d Trytten, D.O. (1982): Wave forces on inclined smooth and rough circular cylinders. Proc. 14th Offshore Technology Conf., Houston, T X , O T C 4227, p p . 731-736.Sarpkaya, T. (1984): Discussion of "Quasi 2-D forces on a vertical cylinder in waves", (paper No. 17671 by P.K. Stansby et al.). J. Waterway, Port, Coastal and Ocean Engineering, 110(1):120-123.Sarpkaya, T. a n d Wilson, J.R. (1984): Pressure distribution on smooth and rough cylinders in harmonic flow. Proc. Ocean Structural Dynamics, Corvallis, OR, 1984, p p . 341-355.Sarpkaya, T. a n d Storm, M. (1985): In-line force on a cylinder translating in oscillatory flow. Applied Ocean Research, 7(4):188-196.Sarpkaya, T. (1986a): Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 165:61-71.Sarpkaya, T. (1986b): In-line a n d transverse forces on smooth a n d rough cylinders in oscillatory flow at high Reynolds numbers. Technical Report No. N P S - 69-86-003, Naval Postgraduate School, Monterey, CA.Sarpkaya, T. (1987): Oscillating flow about smooth a n d rough cylinders. J. Off- shore Mechanics a n d Arctic Engineering, ASME, 109:307-313.Sarpkaya, T . (1990): O n t h e effect of roughness on cylinders. Proc. 9th Offshore Mech. a n d Arctic Engrg. Conf., Feb. 18-22, 1990, Houston, T X , 1(A):47- 55.Sawaragi, T. a n d Nochino, M. (1984): Impact forces of nearly breaking waves on a vertical circular cylinder. Coastal Engineering in J a p a n , 27:249-263.Schewe, G. (1983): O n the force fluctuations acting on a circular cylinder in cross- flow from subcritical u p to transcritical Reynolds numbers. J. Fluid Mech., 133:265-285.
    • 208 Chapter J: Forces on a cylinder in regular wavesSoulsby, R.L., Hamm, L., Klopman, G., Myrhaug, D., Simons, R.R. and Thomas, G.P. (1993): Wave-current interaction within and outside the bottom boundary layer. Coastal Engineering, 21:41-69.Stansby, P.K., Bullock, G.N. and Short, I. (1983): Quasi 2-D forces on a vertical cylinder in waves. J. Waterway, Port, Coastal and Ocean Eng., ASCE, 109(1):128-132.Stansby, P.K. and Smith, P.A. (1991): Viscous forces on a circular cylinder in orbital flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 229:159- 171.Stansby, P.K. (1993): Forces on a circular cylinder in elliptical orbital flows at low Keulegan-Carpenter numbers. Applied Ocean Res., 15:281-292.Stokes, G.G. (1851): On the effect of the internal friction of fluids on the motion of pendulums. Trans. Cambridge Phil. Soc, Vol.9, Part II, pp. 8-106.Sumer, B.M., Jensen, B.L. and Freds0e, J. (1991): Effect of a plane boundary on oscillatory flow around a circular cylinder. J. Fluid Mech., 225:271-300.Sumer, B.M., Jensen, B.L. and Freds0e, J. (1992): Pressure measurements around a pipeline exposed to combined waves and current. Proc. 11th Offshore Mechanics and Arctic Engineering Conf., Calgary, Canada, June 7-11, 1992, V-A:113-121.Tanimoto, K., Takashi, S., Kaneko, T. and Shiota, K. (1986): Impact force of breaking waves on an inclined pile. 5th Int. OMAE Symp., Tokyo, Japan, 1:235-241.Taylor, J.L. (1930): Some hydrodynamical inertia coefficients. Philosophical Mag- azine, Series 7, 9:161-183.T0rum, A. (1989): Wave forces on pile in surface zone. ASCE, J. Waterway, Port, Coastal and Ocean Engineering, 115(4):547-565.Wang, C.Y. (1968): On high-frequency oscillatory viscous flows. J. Fluid Mech., 32:55-68.Weggel, J.R. and Maxwell, W.H.C. (1970): Experimental study of breaking wave pressures. Proc. Offshore Tech. Conf., TX, OTC 1244, pp. 175-188.Wiegel, R.L. (1982): Forces induced by breakers on piles. Proc. 18th Int. Conf. Coastal Engineering, Cape Town, ASCE, New York, pp. 1699-1715.
    • References 209Williamson, C.H.K. (1985): Sinusoidal flow relative to circular cylinders. J. Fluid Mech., Vol. 155, p. 141-174.Wolfram, J. and Theophanatos, A. (1989): T h e loading of heavily roughened cylinders in waves and linear oscillatory flow. Proc. 8th Offshore Mechanics and Arctic Engineering Conf., T h e Hague, March 19-23, 1989, p p . 183-190.Wolfram, J., Javidan, P. and Theophanatos, A. (1989): Vortex shedding and lift forces on heavily roughened cylinders of various aspect ratios in planar oscillatory flow. Proc. 8th Offshore Mechanics and Arctic Engineering Conf., T h e Hague, March 19-23, 1989, pp. 269-278.Yamamoto, T., Nath, J.H. and Slotta, L.S. (1974): Wave forces on cylinders near plane boundary. J. Waterway, Port, Coastal Ocean Engng. Div., ASCE, 100:345-360.Yamamoto, T. and Nath, J.H. (1976): High Reynolds number oscillating flow by cylinders. Proc. 15th Int. Conf. on Coastal Engrg., 111:2321-2340.Yuksel, Y. and Narayanan, R. (1994): Breaking wave forces on horizontal cylinders close to the sea bed. Coastal Engineering, 23:115-133.
    • Chapter 5. Mathematical and numerical treatment of flow around a cylinder T h e mathematical/numerical treatment of flow around cylinders has beenimproved significantly with t h e increasing capacity of computers. This chaptertreats the mathematical/numerical modelling of flow past cylinders; three cate-gories are examined: 1) the methods involving the direct solutions of the Navier-Stokes equations, 2) t h e vortex methods, and 3) t h e methods involving t h e hydro-dynamic stability analysis.5.1 Direct solutions of Navier-Stokes equations T h e direct solution of the complete flow equation is until now restricted onlyto the low Reynolds number case, where the flow is laminar. Numerical solution ofthe N.-S. equation at higher Reynolds number including turbulent features is underway (Spalart and Baldwin (1987) achieved a solution of t h e oscillatory boundarylayer over a plane bed up to Re ~ 10 5 using direct simulation).
    • Direct solutions of Navier-Stokes equations 2115.1.1 Governing equations T h e motion of fluid around a body is governed by t h e Navier-Stokes equa-tions p( ^T + u V U J = - V P + y" V 2 " (5-1)and the continuity equation V -u = 0 (5.2) 2Here u is the velocity vector, p the pressure, y the vector gradient, V the Lapla-cian operator, p t h e fluid density and ft the fluid viscosity. Dots represent thescalar multiplication of two vector quantities (Batchelor, 1967). Past work regarding t h e solution of the Navier-Stokes equations in relationto flow around cylinders are summarized in Table 5.1.5.1.2 The Oseen (1910) and Lamb (1911) solution T h e pioneering work in conjuction with the viscous-fluid flow around bluffbodies dates back as early as 1851; Stokes (1851) treated t h e case of a sphericalbody and determined the flow field around and the drag on the spherical body.He achieved this u n d e r t h e assumption t h a t t h e motion is extremely slow (thecreeping motion) so t h a t Re -C 1. In this case, the inertia forces will be smallcompared with the viscous forces, therefore Eq. 5.1 can be approximated to 0 = - V P + ^V2u (5-3)Stokes obtained a solution to this linear equation and computed the drag, FD, onthe spherical body as Fn 24 CD= = ;Re<<1 (M) U4)^ ^in which CD is the drag coefficient, U t h e velocity of the body and D the diameterof the body. T h e basic ideas behind Stokes analysis is outlined in Example 5.1.
    • 212 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Table 5.1 A partial list of the past work regarding the solution of the two-dimensional Navier-Stokes equations for flow around a cylinder in steady current. Author Re Cylinder Remarks Oseen (1910) i?e< 1 Circular - and Lamb (1911) Thorn (1933) 10 and 20 » - Kawaguti (1953) 40 55 - Apelt (1961) 40 and 44 55 - Fromm & Harlow 15 < Re < 6000 Rectangular For i?e<40 (1963) flow remained steady after the introduction of perturbation to excite vortex shedding Keller k Takami (1966) 2, 4, 10 and 15 Circular - Son & Hanratty (1969) 40, 200 and 500 » No perturbation to excite vortex shedding; only steady-state solutions Dennis & Chang (1970) 5 < Re < 100 55 " Jordan k Fromm (1972) 100, 400 and 1000 55 Vortex shedding is excited by a perturbation
    • Direct solutions of Navier-Stokes equations 213 Table 5.1 continued Author Re Cylinder RemarksDavis k. Moore 100 < Re < 2800 Square No perturbation;(1982) vortex-shedding is excited by round-off errors i) 250 and 1000 Square, 1) " rectangular 2) Effect of angle of attack, effect of shear, effect of aspect ratioBraza, Chassaing 100, 200 and 1000 Circular Vortex shedding& Minh (1986) is excited by a perturbationLecointe & Piquet 140 < Re < 2000 « »(1989) 15 15Braza, Chassaing 2000 < Re < 10000& Minh (1990)Franke, Rodi 40 < Re < 5000 Circular No perturbation;& Schonung (1990) 70 < Re < 300 Square vortex shedding is excited by round- off errorsWang & Dalton 300 < Re < 1000 Circular Vortex shedding is(1991a) excited by a per- turbation. Calcu- lations are extend- ed so as to cover the decelerated- flow nBraza, Nogues 20000 and 30000 -& Persillon (1992)
    • 214 Chapter 5: Mathematical and numerical treatment of flow around a cylinderExample 5.1: D r a g o n a s p h e r e at s m a l l R e y n o l d s n u m b e r T h e sphere is held stationary a n d t h e fluid moves with a velocity U inthe negative direction of the x-axis (Fig. 5.1). T h e spherical coordinate systemis chosen. Only two coordinates, namely, r and 6, will be involved due to theaxisymmetric character of t h e problem Figure 5.1 Definition sketch. Flow around a sphere. By taking t h e divergence of b o t h sides of Eq. 5.3 0 = -V2P + ^V2(V-u) ( 5 -5)and using Eq. 5.2, t h e pressure is found to satisfy the Laplace equation: 2 V p = 0 (5.6) A general solution t o t h e Laplace equation (Eq. 5.6) can be given as aninfinite series of spherical harmonics. However, in the present problem, it turnsout t h a t the previously mentioned infinite series solution is unnecessary, and t h a tthe solution corresponds to a doublet flow p = —-cosf? (5.7)which is known to be a spherical harmonic (Milne-Thomson, 1962, Section 16.1).Here a is a constant. Now the outer boundary conditions d e m a n d t h a t t h e flow approaches to auniform s t r e a m
    • Direct solutions of NavieT-Stokes equations 215 j/> — — -Ur2 > sin 2 8 as r —• oo (5.8)(Milne-Thomson, 1962, Section 15.22). Hence a general expression for the streamfunction can be sought in t h e following form f = - f(r) sin2 6 (5.9)in which / is an unknown function. Now consider the i-component of the equation of motion, Eq. 5.3, ^ = ^V2« (5.10)and insert the following identities into t h e above equation f- = f cos6--^ sine (seeFig.5.1) (5.11) Ox Or r 38 u = vr cos 6 — vg sin 9 (see Fig.5.1) (5-12) 2 1 d ( 2 d 1 d (d . n ,c . v r + 2 sme (5 13 -^{ o-r) ^ ^eo~e{de ) - > (the Laplace operator in spherical polar coordinates) and v vr = ^~"35" e = —^~2^- ( 5 -14) r sin e r o9 r sin 8 Or (in spherical polar coordinates)in which p is given by Eq. 5.7 and t/> is given by Eq. 5.9. This yields — cos^ 9 sin*1 e = r r = /i [ - ! £ + 2 / " ] cos 2 0 - ^ [ ^ - i ( - r / " ] sin 2 0 (5.15)By setting t h e factors in front of sin 2 9 and cos 2 6 equal to zero, t h e following twoordinary differential equations are obtained: 2 r / " - 2 / = - r and r3 f" - 2rf + 4/ = - — (5.16)which both have the solution -*- =+£ <"•">
    • 216 Chapter 5: Mathematical and numerical treatment of flow around a cylinderin which ft and ft are arbitrary constants. From the outer boundary condition, namely / —• hUr2, Eqs. 5.8 and 5.9, 3^-U (5.18)On the surface of t h e sphere, on the other hand, vr = vg = 0 which, from Eq. 5.14reads /(r0)=0 and /(r0) = 0 (5.19)in which t h e constans a and ft are found as follows 3 1 a = -fiUr0 and ft = -t/r„ (5.20)The velocity components are therefore 3ro_ I / ^ V vT = U cos t 2r 2Vr / 3 r o _ 1-/^o 3 vg = U sin 0 (5.21) 4 r 4V r / T h e force on t h e sphere due to pressure will be (using Eqs. 5.7 and 5.20) Fp = — 27rr0 / psin6 cos 8 d8 = 2irnroU (5.22) oT h e force on the sphere due to friction, on the other hand, will be 7T Ff = -27TT, / Trs sin 2 6 d9 = 4Tr/j,roU (5.23)in which r r S is calculated from r r e = -[idve/dr, yielding 3v» 3 U Tre = -V-^- = --/*—sine (5.24) or 2 roHence, t h e total force from Eqs. 5.23 and 5.24 will be F = Fp + Ff = 6*fir0U (5.25)which, in terms of drag coefficient, can be written as in Eq. 5.4. As a final remark, the solution (Eq. 5.21) is self-consistent at positions nearthe sphere in the sense that the inertia forces are small compared with t h e viscousforces, justifying the creeping motion assumption leading to Eq. 5.3. However,the inertia forces corresponding to the solution (5.21) become comparable with
    • Direct solutions of Navier-Stokes equations 217viscous forces at distances from the sphere of order r0/R (Batchelor, 1967, p.232). (The solution is clearly not valid at such large distances). This is calledOseens paradox. We shall r e t u r n to this problem in the next example.E x a m p l e 5.2: D r a g o n a circular c y l i n d e r at s m a l l R e y n o l d s n u m b e r A solution to Eqs. 5.2 and 5.3 may be sought for a circular cylinder in thesame way as for a sphere. T h e pressure is given by the following equation (in place of Eq. 5.7) p=--cos0 (5.26) rin which (r, 8) are the polar coordinates (Fig. 4.3). T h e analogue of Eq. 5.9 is 0 = -/(r)sin6l (5.27)T h e differential equations satisfied by the function / (the analogues of Eqs. 5.16) 2 f" + rf -f = -r and r3 f" + r2 f" - 2rf + 2f =--r (5.28) H fiwhich both have the solution / = i - r l n r - f t r - ^ (5.29)in which /3 and /?2 are arbitrary constants. On the surface of the cylinder, vr = vg = 0, i.e., wr = - - | £ = 0 and ve = ^-=0 (5.30) r off Oror, from Eq. 5.27 /(r0) = 0 and f(r0) = 0 (5.31)From the latter two equations, the constants fix and /?2 are found as follows 1 rv 1 cv cvTn h = -A-+~-^r0 and fo =-~r± (5-32) 4 ft 2n 4/iHence, the velocity components vr and VQ are vr = — f c o s 8 and vg = — fsin8 (5.33) r
    • 218 Chapter 5: Mathematical and numerical treatment of flow around a cylinderin which lnr ~~ (2 +lnr °)r + 2r<>; (5.34) 2 fj. T h e force on the cylinder due to pressure will be (using Eq. 5.26) Fp = - I p(r0d6) cosfl = na (5.35) 0and the force due to friction will be 2?r Z7T F,> = -- / Tr9(r0d6) sin0 = wa (5.36) 0in which rrg is calculated from rrg = fidyg/dr, giving dv$ a . „ , Tr9 = /x—— = sm0 (5.37)From Eqs. 5.35 and 5.36, the total force on the cylinder will be F = Fp + Ff = 2na (5.38) T h e remaining arbitrary constant a has to be determined from the outerboundary condition. However, no choice of a will make u go to t h e constant valuecorresponding to the undisturbed flow, as r — 00, since / diverges as l n r when >r is large (Eq. 5.34). It can be shown t h a t the inertia force becomes comparablewith the viscous force at large distances from the cylinder, and the solution (5.34)is thus not a self-consistent approximation to the flow field at large values ofr (Oseens paradox). Clearly some approximation to the equation of motion atlarge r is needed, and Eq. 5.34 must m a t c h with t h e solution of this approximateequation at large distances from the cylinder. It can be shown t h a t this approximation to the equation of motion is -pU • v u = - V P + f V2 u (5-39)This, together with t h e equation of continuity, Eq. 5.2, are known as t h e Oseenequations (Oseen, 1910). T h e calculations due to Lamb (1911) show t h a t Eq. 5.39has a solution which, near the cylinder, approximates to the solution (Eq. 5.34)provided t h a t t h e constant in Eq. 5.34 is chosen as (Batchelor, 1967, p. 246) (5 40) °=w?m -Thus the drag coefficient, from Eqs. 5.38 and 5.40 will be
    • Direct solutions of Navier-Stokes equations 219 C ° = ReHlA/Re) * « * ^This relation is in good agreement with experiments for values of Re up to about0.5 (Fig. 2.7).5.1.3 Numerical solutions The N.-S. equations and the continuity equation, Eqs. 5.1 and 5.2, for atwo-dimensional flow in a Carterian co-ordinate system are du du du d(p/p) (d2u d2u +u +v + =v (5 42) m irx Ty ^ W + w)> - 2 2 dv_ dv dv d(p/p) _ (d v d v (KAO dt dx dy dy dx2 dy2) du dv ,, ,,,in which u a n d v are t h e components of velocity along t h e x a n d y directions,respectively. It is more convenient t o write t h e N . - S . equations in terms of t h e streamfunction, rjj, a n d t h e vorticity function, OJ, defined by u = £ (5.45) v= -§* (5.46) dv du , ^ Y x d y ^ The continuity equation (Eq. 5.44) is satisfied automatically by Eqs. 5.45and 5.46. Regarding the N.-S. equation (Eqs. 5.42 and 5.44), eliminating thepressure from these equations and making use of Eqs. 5.45 - 5.47, the followingequation is obtained doj du du> fd2w d2u> +U +V = 2 + (5 48) Tt Tx lTy -dz- w ) This equation is known as the vorticity-transport equation.
    • 220 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Inserting Eqs. 5.45 and 5.46 into Eq. 5.47, on the other hand, the so-calledP o i s s o n e q u a t i o n is obtained d2j> <92V> (5.49) dx2 dy2Eqs. 5.45-5.49, or their polar co-ordinate counterparts, constitute the basic equa-tions used in a numerical solution of N . - S . equations. These equations are to besolved with the boundary conditions on the cylinder surface and at a boundaryfar away from the cylinder (the outer boundary). T h e requirements are: On thecylinder surface the no-slip and impermeability conditions must be satisfied while,at the outer boundary, the velocity components must be identical to those of theundisturbed flow. Wall Sx 8y —?— rr r± Rectangular cylinder Wall Figure 5.2 Portion of computational region showing finite-difference mesh and its relation to solid boundaries. T h e basic principles of such a numerical study may be described by thefollowing example, which is taken from the work by Fromm and Harlow (1963),(see Table 5.1). A rectangular cylinder with a large aspect ratio is impulsivelyaccelerated to a constant velocity in a channel of finite width. A finite-differencemesh of cells of sides Sx and Sy, dividing the spatial region of interest in themanner shown in Fig. 5.2, is introduced. In this way, the continuous flow field can be described by a finite numberof quantities. T h e basic steps involved in advancing the solution from time t totime t + 6t are as follows:
    • Direct solutions of Navier-Stokes equations 221 Flow 1 IB Body Hi /* 11 2 • 3 [••ffi-M O u t e r edge of computational domain Figure 5.3 The pressure distribution is determined by numerically integrat- ing the momentum equations (Eqs. 5.50 and 5.51) over 1234... 1. At t h e beginning, all required quantites are available in t h e computer mem- ory. 2. For each "vorticity" point, a new value of to is found by use of a finite- difference approximation of Eq. 5.48. 3. For each "stream-function" point, a new value of if> is found from a finite- difference approximation of Eq. 5.49. (The method of solution involves a succession of iterations). 4. Implementing Eqs. 5.45 and 5.46, t h e new components of velocity are found, where care is taken in the entire procedure t h a t the results are consistent with the finite-difference form of Eq. 5.47. 5. Given t h e velocity and the vorticity field, the pressure is then calculated, using the following equations: On y = constant lines: p 2 p 2 B du f f du / A —dx A + / vudx — / v— A (5.50)in which A and B are two points on the j/-constant line.
    • 222 Chapter 5: Mathematical and numerical treatment of flow around a cylinder On x = constant lines: p 2 /» 2 w i< u dy uu,dy + 3u> / at ~ J / 9x " (5.51)This equation is a version of t h e energy equation in a viscous fluid (they can easilybe obtained from Eqs. 5.42, 5.43, 5.45, 5.46, 5.47 and 5.49). •?. v.i a) —-Ji J~~ir*?- v "••* •. >•;•:". -:v-; W/.V b) Figure 5.4 Snap shot of flow around a rectangular cylinder, a) Numeri- cal results by solution of the 2D N.-S. equations Re = 6000. The cylinder-height-to-channel-width ratio ( = D/H) = 1/6. b) Experiment. Fromm and Harlow (1963) with permission - see Credits. To get t h e pressure on t h e b o d y surface, Eq. 5.50 is first applied on line 12(Fig. 5.3), t h e n Eq. 5.51 on line 23, then Eq. 5.50 on line 34 and so on. To gett h e wall shear stress on t h e body surface
    • Direct solutions of Navier-Stokes equations 22S T = fi— (on horizontal lines) (5.52) dyand dv r = / x — (on vertical lines) (5.53) oxmust be applied. Integrating t h e pressure and wall shear stress distributionsaround the cylinder surface gives the instantaneous resultant force on the cylinder. 50 tU/D 50 tU/D Figure 5.5 Time series of drag and lift coefficients for a circular cylinder ob- tained numerically from the solution of the 2D N. - S . equations in steady current. Re = 200. Braza et al. (1986). Although t h e underlying principles of a numerical solution of the N . - S .equations for flow around a cylinder may appear to be quite straightforward, thereare numerous details involved in t h e solution procedure to ensure t h a t the solutionis b o t h stable and sufficiently accurate: these details are related to various aspectsof t h e problem such as t h e b o u n d a r y conditions; t h e choice of Sx, Sy and, St; t h estability of t h e finite-difference equations; t h e introduction of a perturbation toinitiate t h e vortex shedding within a short time interval; and so on. Also, the
    • 224 Chapter 5: Mathematical and numerical treatment of flow around a cylinderfinite-difference scheme used in the solution of the equations may have a directinfluence on the end results (Borthwick, 1986). Fig. 5.4a gives a snapshot of t h e flow obtained in the study of F r o m m andHarlow (1963) while Fig. 5.4b gives t h a t from an actual experiment. As seen, thenumerical results reveal t h e main features of t h e flow quite well. Fig. 5.5 illustrates t h e time series of t h e drag and lift coefficients for acircular cylinder obtained numerically by solving t h e N . - S . equations for Re = 200(Braza et al., 1986). T h e forces reach a steady s t a t e with periodic oscillations aftera transient time interval. T h e vortex shedding is excited in Braza et al.s studyby a physical perturbation imposed numerically. Tl!M| i i i Mini i i i i II ii| i i i 111 I I St Williamson (1989) 1 FvnPrtmental Roshko (1961) I Experimental • J o r d a n & Fromm (1972) 1 0.4 Braza e t a l . (1986) „ J , Braza et al. (1990) Numerical + Braza et al. (1992) J 0.3 — " " 0.2 - ,^z--o---g--*"*--*--*--+ + 0.1 - 1 i ml 1 III 1 1 1 1 1 1 III 1 1 1 1 1 M l 40 10" 10° 10 10° Re Figure 5.6 Strouhal number for a circular cylinder in steady current. Nu- merical results are from the solutions of the 2D N.-S. equations. Fig. 5.6 compares the numerically obtained results regarding t h e Strouhalnumber with the experiments in the case of circular cylinder. Likewise, Fig. 5.7compares the mean drag coefficient obtained numerically with t h e experiments.T h e numerical d a t a in t h e figures are all from the solutions of the 2D N . - S . equa-tions. T h e agreement between t h e numerical results and the experiments is quitegood.
    • Direct solutions of Navier-Stokes equations 225 1—rrn—i I Ml I ill i I l l| I l I II i I ill—I I I I Trltton 11959) Wleselsberger (See Schllchtlng (1979)) } mental Experi- — — Oseen - Lamb relation, Eq. 5.41 Thorn (1933) o Kawagutl (1953) 100 X Apelt (1961) Keller & Takaml (1966) Jordan & Fromm (1972) ) Numerical D Braza et al. (1986) s Braza et al. (1990) Braza et al. (1992) 10 0.1 _ 1 I I I I I i l d_ I I I I I t I 1 II 10 10 10 10* 10° 10" 10" 10 Re F i g u r e 5.7 Mean d r a g coefficient for a circular cylinder in s t e a d y cur- r e n t . Numerical results are from t h e solutions of t h e 2D N . - S . equations.T u r b u l e n t flow Until now, the numerical solution of the two-dimensional N . - S . equationshas been discussed. It is known, however, t h a t t h e flow around t h e cylinder is two-dimensional only when Re < 200. For larger Re numbers, the vortex sheddingoccurs in cells and therefore the flow is three-dimensional (Fig. 1.26 and Sections1.1 and 1.2.2). Hence, for such Re numbers, t h e 2D N . - S . solution is only anapproximation. Although the 2D N . - S . solutions give fairly good agreement withthe measurements with regard to the gross-flow parameters (Figs. 5.4, 5.6 and5.7), this is not so, however, for the lift force for instance; see Fig. 5.8. As seenfrom Fig. 5.8, the lift force is grossly overpredicted. This may b e due partly to t h e2D computations: in the real flow, the presence of cells implies t h a t the lift doesnot take place concurrently along the whole length of t h e cylinder, thus reducingthe average lift. (Note t h a t the two values plotted in Fig. 5.8 were obtained, usingtwo different grid sizes in Braza et al.s (1990) study). For Re numbers smaller t h a n 3 x 10 5 (but larger t h a n 300), the flow isturbulent in the wake (Fig. 1.1). W h e n Re is increased further, turbulence beginsto spread into the boundary layer (Fig. 1.1 g-i). So, in this situation, the instan-
    • 226 Chapter 5: Mathematical and numerical treatment of flow around a cylinder l I I F i I rn "1 T-TT 1.0 h —— Schewe : Experimental (1983) • Braza et ai. : Numerical (1990) 0.8 0.6 0.4 - 0.2 0.0 10 10 10 10 Re Figure 5.8 R.m.s. value of oscillating lift in steady current. Numerical results are from the solutions of the two-dimensional N.-S. equations.taneous flow is three-dimensional not only in the wake b u t also in boundary layeritself. It is possible to carry out 3D computations where the 3D N . - S . equationsare solved numerically. This method, called the direct numerical simulation of N . -S. equations, is presently feasible only for relatively small Re numbers; for largeRe numbers, t h e scales of t h e dissipative p a r t of turbulent motion are so smallthat this kind of small scale motion can not be resolved in a numerical calculation(the number of grid points required to resolve this motion increases approximatelywith Re3) (Rodi, 1992). We shall return to the issue of 3D computations later inSection 5.2. It is clear from the preceding discussion t h a t , for relatively large Re numbers(where the flow in the cylinder boundary layer is turbulent), the direct numericalsimulation of the N . - S . equations is not feasible. Similar arguments can be rea-soned also for the case of rough-surface cylinders. So, in such situations, it maybe desirable to solve the flow equations in such a way t h a t the turbulence effectsare modelled by use of a turbulence model such as an eddy-viscosity model ora Reynolds-stress-equation model or a large-eddy simulation model. An accountof such a model (Justesen, 1990) is given in the next section. A review of theturbulence models as applied to flow past bluff bodies in steady current has beengiven by Rodi (1992).
    • Direct solutions of Navier-Stokes equations 2275.1.4 A p p l i c a t i o n t o o s c i l l a t o r y flow Stokes (1851) was the first to develop an analytical solution for the 2DN . - S . equations for t h e case of a cylindrical body oscillating sinusoidally in aviscous fluid, as has already been pointed out in conjunction with the asymptotictheory described in Example 4.3. (Recall t h a t t h e results of t h e asymptotic theoryin Example 4.3 are t h e same as t h e Stokes theory to 0{(Re/RC)1/2]). Wang(1968) later extended Stokes analysis to 0[(Re/KC)-3/2]. Figure 5.9 Computed vorticity contours due to N.-S. solution for KC = 8 and j3 = 196. Four instances are shown: (a-d) ^7r; 7r; |7r and 2ir respectively. Justesen (1991) with permission - see Credits.
    • 228 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Regarding the numerical treatment of the problem, the equations which areto be integrated numerically are t h e same as those given in the previous section,namely, the vorticity transport equation (Eq. 5.48) and the Poisson equation (Eq.5.49). T h e outer flow velocity is now a periodic function of time: U = Um sin(u;£) (5.54) -i 1 1 1 1 r n 1 r a) in m»< "AA -I L J I I I L 10 15 20 25 30 KC i 1" i i i i i 1 1 1 . b) 2 ^"HV i - A A A i 1 i i 1 i 1 1 I 10 15 20 25 30 KC Figure 5.10 Circles: Computed in-line force coefficients due to N.-S. solu- tion of Justesen (1991). /3 = 196; Triangles: Experiments by Obasajuetal. (1988). (a) Drag coefficient; (b) inertia coefficient. B a b a and Miyata (1987) were the first to a t t e m p t at solving the N . - S .equations for a sinusoidal flow. They presented two calculations; one for thecombination of KC = 5 and Re = 1000, and the other for KC = 7 and Re = 700.In b o t h calculations, the wake was symmetric in contrast to observations (Fig.
    • Direct solutions of Navier-Stokes equations 2293.16). Murashige, Hinatsu and Kinoshita (1989) have m a d e similar calculationsfor three KC numbers, KC = 5,7 and 10, for Re numbers around 10 4 . In thelatter work, t h e flow was perturbed, to trigger asymmetry for relatively smallKC numbers and eventually to excite vortex shedding for larger KC numbers.Apparently, these authors were able t o obtain t h e transverse-vortex street regime(Figs. 3.6a and 3.7) for KC = 10. Later Wang and Dalton (1991b) m a d e similarcalculations for KC ranging from 1 to 12 and Re ranging from 100 t o 3000. T h elatter authors reported their results also in Zhang, Dalton and Wang (1991). Justesen (1991) has m a d e an extensive study of oscillating flow around acircular cylinder, solving t h e N . - S . equations numerically for a wide range of KC,namely 0 < KC < 26, and for three values of /?(= Re/KC) in the range 196-1035. Fig. 5.9 shows the computed vorticity contours for KC = 8 and /? =196. T h e presence of t h e transverse-vortex street is quite evident. Justesen alsocomputed the conventional force coefficients (for all three /9 values). Figs. 5.10and 5.11 compare Justesens numerical results for /? = 196 with t h e results fromexperiments. z.u I I I I I I I < i i 1.5 Lrins 1.0 a a B• •a a a =6 o • 0.5 n a W DQ «J I i i • i I i I i 0 5 10 15 20 25 30 KC Figure 5.11 Lift. /3 = 196. Circles: Justesens N.-S. solution (1991). Squares: Experiments by Maull and Milliner (1978) for /3 = 200. Lift force in Maull and Millners experiments was measured by strain gauges and represents the force on the total length of the cylinder, L, the ratio L/D being approximately 18. In Justesens calculations, the Reynolds number was kept rather small (/? =196) such t h a t t h e effect of transition and turbulence remain as small as possible.For t h a t reason, the computations were stopped at KC = 26. Although it may beargued t h a t even KC = 26 may b e too high for t h e turbulent effects to be negligible(see Fig. 3.15), the agreement between t h e numerical results and t h e experimentald a t a is rather good with regard to the in-line coefficients (Fig. 5.10). This may
    • 2S0 Chapter 5: Mathematical and numerical treatment of flow around a cylinderbe due to the fact t h a t the flow is turbulent only in p a r t s of t h e oscillation cycleor in t h e wake such t h a t the boundary layer separation is predominantly laminar.Regarding t h e lift coefficients (Fig. 5.11), there is some discrepancy. Apparently,the numerical predictions of lift agree extremely well with t h e experimental d a t a atKC numbers KC = 10, 18 and 26 where large spanwise correlations are measured(Fig. 3.28). However, for KC values where t h e spanwise correlation is small theflow is strongly three-dimensional and therefore it is expected t h a t a 2-D model isnot able to handle such cases. This is quite evident from Fig. 5.11 (c.f. Fig. 3.28).Finally, Fig. 5.12 presents Justesens results regarding CD and CM coefficientsobtained for the highest /3-value, namely 0 = 1035. _ -D 2.0 0.1 0.2 0.4 0.6 1.0 10 KC 2.5 -i 1—i—i—i i i i i b) r 2.0 "# * * « * * * « l # t # n + + 1 1.5 _i i i i 11111 _i i i i i i_ 1.0 0.1 0.2 0.4 0.6 1.0 2 4 6 10 KC Figure 5.12 (a) Drag coefficient; (b) Inertia coefficient. Circles: Com- puted in-line force coefficients due to N.-S. solution of Juste- sen (1991). f) = 1035; Squares: Discrete vortex method by Stansby and Smith (1989); Crosses: Experiments by Sarpkaya (1986); —, asymptotic theory (Eqs. 4.57 and 4.58).
    • Direct solutions of Navier-Stokes equations 231 —r- n—r 1 1 — " I " 1—i—r 1 i 1 «A4A****K* 2 - A * * • A | A ^ • *** • • • • A 1 1 1 1 i . l . l , 1 1 L L I 0 2 4 6 8 10 12 14 16 KC T—i—I—i—|—i—l—I—|—r "* ^5 ULt * i - **t »*i1M|A. J t I . I |_ o0 i 2 i i 4 6 8 10 12 14 16 KC Figure 5.13 Circles: Computed in-line force coefficients due to turbulent N.- S. solution of Justesen (1990). k3/D = 4.8 X 1 0 " 2 . Triangles: Experiments by Justesen (1989). Justesen (1990) treated also t h e case of turbulent flow where the turbulenceeffects were modelled by use of a one-equation turbulence model for a rough cylin-der. The equations are essentially the same as in the case of laminar flow, namelythe vorticity transport equation and the Poisson equation. The only difference isthat, in the present case, the vorticity transport equation includes also the so-calledturbulent viscosity, vT. VT is modelled by a one-equation model. This presumablyadds one more equation (namely, the equation for turbulent energy) to the set ofequations which is to be solved. Justesen (1990) carried out his calculations forKC numbers up to KC = 10 for a cylinder roughness of ks/D = 48 x 10~ 3 . Fig.5.13 compares his numerical results with the results of experiments reported inJustesen (1989).
    • 232 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Uc/Um: f>?^Ii3P^Si*j[^ 0.6 2 <5>^S ^ " ^ ^ ^ ^ Figure 5.14 Streaklines obtained from N.-S. solutions. For combined oscil- latory-flow and current environment KC = 4. /? = 200. Sarpkaya et al. (1992) with permission - see Credits.
    • Discrete vortex methods 233 T h e N . - S . solutions have been obtained also for t h e case of combined os-cillatory flow and current (Sarpkaya, Putzig, Gordon, Wang and Dalton, 1992).T h e calculations were carried out for KC = 1-6 with fi = 200(.Re = 800-1200) forvarious values of Uc/Um = 0-1.2 in which Uc is t h e current velocity. T h e resultshave revealed t h e existence of a wake feature in t h e interval Uc/Um = 0.6-0.8 forKC = 4 (Fig. 5.14) different from those in steady currents a n d in oscillatory flows.Furthermore, Sarpkaya et al. obtained reasonable agreement with t h e experimentsregarding t h e in-line coefficients for KC = 4 - 6 . Recently Badr et al. (1995) have reported t h e results of a numerical solutionto t h e N.-S. equations for Re = 10 3 , KC = 2 and 4, and for Re = 10 4 a n d KC = 2.As mentioned in C h a p t e r 3, their results revealed t h e presence of steady streamingpatterns (shown in Fig. 3.38) even in t h e case of separated flows.5.2 Discrete vortex methods In practice, large difficulties are encountered for solving t h e N . - S . equationsusing t h e finite-difference or finite-element methods. One of t h e major difficultiesis t h a t t h e number of grid points (therefore, t h e amount of computation) requiredto obtain a solution increases with increasing Reynolds number, a n d m a y becomeprohibitive at large Reynolds numbes, as mentioned earlier. It is therefore ofinterest t o develop a grid-free (or almost grid-free) numerical method. A simplemethod offering an alternative t o t h e finite-difference method is t h e discrete vortexmethod. T h e equations t o be solved are exactly t h e same as in t h e preceding section,namely t h e vorticity-transport equation (Eq. 5.48) and t h e Poisson equation (Eq.5.49): duj dw dto (d2ui d2 *u> +u +v =v + dt di d^ {d^ W] (5 55) - d2i> d2ij) _ (5.56) dx2 dy2In principle, t h e only difference between t h e vortex methods a n d t h e finite-difference methods is that t h e solution t o t h e vorticity-transport equation (Eq.5.55) in case of vortex methods is obtained through a numerical simulation ofconvective diffusion of discrete vortices generated on t h e cylinder boundary (thenumerical simulation of vorticity t r a n s p o r t ) . In t h e following, attention will be concentrated first on t h e simulation ofvorticity transport. This will follow by t h e description of t h e underlying principlesof t h e vortex method as applied t o flow around a cylinder. T h e section endswith illustration of several examples selected from t h e literature, covering b o t hthe steady current and oscillatory-flow situations.
    • 234 Chapter 5: Mathematical and numerical treatment of flow around a cylinder5.2.1 Numerical simulation of vorticity transport There is an analogy between the convective diffusion of any passive quantitysuch as concentration (or temperature in the case of heat transfer) and the trans-port of vorticity. Both processes are governed by the same differential equation.This is seen in Table 5.2 where other elements of the analogy are also indicated.In the table, C is the concentration of the passive quantity and K is the diffusioncoefficient (C and K in the case of heat transfer are the temperature and thethermal conductivity, respectively, Crank (1975)). Table 5.2 Analogy between the convective diffusion of passive quantity and that of vorticity. Convective diffusion of a passive Transport of vorticity quantity such as mass or heat C: Concentration (or temperature) u>: Vorticity Convective diffusion equation: (5.57) Vorticity transport equation: (5.58) + u v A + at dx + dy - ^ dx* Sy» )Standard deviation of particle position: Standard deviation of vortex position: y/r* = J2K St (5.59) Vr« = /2v St (5.60) Lagrangian simulation with particles: Lagrangian simulation with vortices: C = N/A (5.61) cj = VIA (5.62)Numerical simulation of convective diffusion An alternative to solving the convective diffusion equation (Eq. 5.57 in Ta-ble 5.2) is the Lagrangian simulation of convective diffusion process by a random-walk model. This method has been developed over the last decades (Bugliarello(1971), Sullivan (1971) among others) and is now a powerful numerical tool usedin the problems related to diffusion of mass in flow environments. The aforementioned simulation may be described by the following simpleexample. Consider the diffusion of mass from a continuous point source (Fig.
    • Discrete vortex methods 2S5 Mesh element r Continuous point source • • Figure 5.15 Diffusion of passive quantity from a point source.5.15). T h e diffusing mass in the example can be considered as a cloud of largenumber of "particles". E a c h particle actually follows two basic steps, namely 1) a convective step determined by the velocity of t h e field correspondingto the position of the particle, and 2) a r a n d o m diffusive step (Fig. 5.16). T h e m a g n i t u d e and the directionof t h e r a n d o m diffusive step is selected from a Gaussian process with a s t a n d a r ddeviation set equal to y2K St in which St is the small time interval during whichthe particle takes its step (Eq. 5.59 in Table 5.2). In t h e simulation, many suchparticles released from the source point are followed as they travel through thestatistical field variables. T h e concentration, C, can t h e n be calculated, in principle, from t h e numberof particles found in a mesh element by C = N/A (Eq. 5.61 in Table 5.2) in whichN is the number of particles in the mesh element and A is t h e area of t h e samemesh element (Fig. 5.15). It can be shown t h a t the concentration obtained in thisway (for large number of particles) is equivalent to t h a t found from the solutionof the convective diffusion equation (Eq. 5.57 in Table 5.2). Diffusive /"^ step / L i+i / Convective step Figure 5.16 Random walk of a particle.
    • 2S6 Chapter 5: Mathematical and numerical treatment of flow around a cylinderN u m e r i c a l s i m u l a t i o n o f c o n v e c t i v e diffusion o f v o r t i c i t y In the case of vorticity transport, the diffusing "mass" of vorticity maybe considered as a cloud of large number of vortex "particles", analogous to thediffusion of particles described in the previous section. T h e vortex "particle" maybe termed t h e vortex blob or the discrete vortex. Obviously, these discrete vorticesmust be generated on the boundaries, each vortex being assigned with a certainstrength and a direction of rotation. In t h e case of a cylinder the vortex generationtakes place on the surface of the cylinder, Fig. 5.17. Mesh element Figure 5.17 Discrete vortices released from the cylinder surface. As in the case of diffusing passive particles described in the preceding sec-tion, the discrete vortices introduced into the flow from the boundaries follow twobasic steps: a convective step and a diffusive step (Fig. 5.16). T h e convectivestep is determined by t h e velocity of the field corresponding to t h e position of thediscrete vortex, while the diffusive step is selected from a Gaussian process witha s t a n d a r d deviation equal to /2v St. (Recall the analogy between the diffusioncoefficient K and t h e kinematic viscosity v, Table 5.2. T h e diffusion here cor-responds to molecular (Brownian) diffusion; in the case of turbulent flow, v hasto be replaced by v + ux, where VT is the turbulence viscosity, to simulate theturbulent diffusion, see Section 5.2.3). Many such vortices are followed, and thevorticity, w, can, in principle, be calculated by w = T/A (Eq. 5.62 in Table 5.2) inwhich T is the sum of the strengths of t h e vortices found in a mesh element (Fig.5.17) and A is t h e area of the mesh element itself. Finally it should be noted t h a t the aforementioned scheme was shown toconverge to the solution of the N . - S . equations (Chorin, Hughes, McCracken andMarsden, 1978)
    • Discrete vortex methods 2375.2.2 P r o c e d u r e used in t h e i m p l e m e n t a t i o n of discrete vortex method As has already been mentioned, t h e principal idea behind t h e discrete vortexmethod is to achieve t h e solution of t h e vorticity-transport equation (Eq. 5.55)through the numerical simulation of vorticity transport. For this, the followingprocedure is used. Figure 5.18 The vortex-induced velocity at the surface of the cylinder to can- cel the existing velocity so that the no-slip boundary condition can be fulfilled on the surface at that particular location. 1. First use the potential flow solution and work out t h e velocity on thecylinder surface. (To avoid numerical difficulties, a gradual (timewise) increase inthe velocity to the value U may be contemplated in the computations). 2. Introduce discrete vortices just above the cylinder surface. For this,determine t h e strengths of these vortices such t h a t t h e no-slip condition is satisfiedon the surface. For example, for the vortex which will be introduced at the topedge of the cylinder at the initial instant (Fig. 5.18), the strength of the vortexshould be r = 2TT 8r (2U) (5.63)and the direction of rotation should be clock-wise, so t h a t t h e velocity just at t h a tpoint on the cylinder surface would be zero:
    • Chapter 5: Mathematical and numerical treatment of flow around a cylinder +2U + (-?L)=+2u-2u=o (5-64) from the from the potential flow solution introduced vortex(Introducing these vortices can be refined. It can be either taken at t h e first meshpoint or distributed over several mesh points, using the boundary-layer theory.) 3. Move t h e vortices according t o t h e random-walk model described in theprevious section. 4. Distribute t h e strengths of vortices on t h e mesh according to a specifiedscheme. For example, according to a weighting scheme which is widely used insimulation studies, a vortex located at Point P in t h e mesh element illustrated inFig. 5.19 generates vorticity at Point i: AT LJi = ~i? i = 1>234 ( 5 - 65 )in which A is t h e total area of the mesh and Ai are t h e areas indicated in Fig.5.19 3 4 A2 Ai •: [• P A4 A3 1 2 Figure 5.19 Vorticity values at the mesh points 1, 2, 3 and 4, caused by the vortex at Point P , are calculated according to the scheme in Eq. 5.65. 5. Given the vorticity values at the mesh points, solve the Poisson equation(Eq. 5.56) numerically and obtain the new velocity components at the mesh pointsby
    • Discrete vortex methods 2S9 cty (5.66) and dy dx 6. Restore the no-slip boundary condition at the surface of the cylinder byintroducing a new set of discrete vortices at the cylinder b o u n d a r y (Fig. 5.20) andrepeat the steps 3 to 6. ,Q O O O Newly created vortices Figure 5.20 Vortex creation to satisfy the no-slip condition. 7. At any time t, given (a) the position of the i th vortex in terms of polarcoordinates r;, 9i (Fig. 5.21), (b) the velocity components t h a t the i th vortexexperiences in the x and y directions, namely U{, Vi, and (c) the vortex strength,Yi, corresponding to the i th vortex, the force components Fx and Fy on thecylinder may be calculated using the following expressions ;sin(20j) - U j c o s ( 2 0 ; ) E* + 2FZ (5.67) Vi sin(2#;) + Ui cos(2#;) •E* 7i + 2F9 (5.68)in which N is the total number of vortices, and FX3 and Fys are t h e force due tosurface shear stress (skin friction) in x and y directions, respectively. T h e skinfriction force is obtained from the surface shear stress which is actually propor-tional to the surface vorticity. T h e quantities appearing in Eqs. 5.67 and 5.68 areobtained by having (a) the position of discrete vortices, (r;, #;), (b) the velocityof discrete vortices, (u;, V{) and (c) the vortex strength, I V T h e method is due toQuartapelle and Napolitano (1983). It reduces to the aforementioned convenientform for the case of a circular cylinder (Stansby and Slaouti, 1993). These are t h e typical steps taken in t h e implementation of the discretevortex method. There are, however, numerous details which need to be taken into
    • &J,0 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Flow x Figure 5.21 Definition sketch for the force calculation in the discrete vortex method.account, such as the choice of the time step 6t in the random-walk simulation, thenumber of vortices introduced per time step, the extent of the mesh (or meshes),the mesh size and so on. (Smith and Stansby, 1988). T h e vortex method was originally proposed by Rosenhead (1931) and fur-ther developed in recent time by Chorin (1973 and 1978). In t h e version t h a tChorin presented, the velocity is calculated by directly summing the influence ofall the other vortices. This may be computationally prohibitive, since there arevery many vortices (O(10 4 )) in the flow. To avoid this, the so-called v o r t e x - i n -cell (or c l o u d - i n - c e l l ) method has been devised (Christiansen (1973) and Baker(1979)). In this method, the contribution of each vortex to t h e vorticity at themesh points is calculated (in the manner as described in Step 4 above) and then thevelocity is obtained by solving the Poisson equation (Step 5 above). Therefore, thedisadvantage of the method requiring a large number of vortices is compensatedby this kind of efficient vortex handling. T h e vortex methods where vorticity is created only at separation point havealso been developed. In this case, the method requires knowledge of separationlocations and therefore these methods may be suitable for bodies with sharp edges. T h e advantages of vortex methods over the other methods to solve the N . - S .equations may be summarized as follows: 1) First of all, the inviscid theory couldbe employed (Step 3 above); 2) the numerical diffusion problems associated withthe vorticity gradient terms in Eulerian schemes are to a large degree avoided; 3)
    • Discrete vortex methods 241there are no zone assumptions which could, for instance, require matching of anouter flow to an inner flow; and finally 4) t h e method is relatively stable and wellsuited to vectorisation on supercomputers (Stansby and Isaacson, 1987). A detailed review of the vortex methods has been given by Leonard (1980)and Sarpkaya (1989).S m a l l .Re-number s i m u l a t i o n b y t h e d i s c r e t e v o r t e x m e t h o d T h e vortex shedding is two-dimensional in the range 40 < Re < 200 (Sec-tion 1.1). Therefore this range of Re number would offer the possibility of trueapplication of the method, since no three-dimensionality is present. Stansby andSlaouti (1993) did computations of the flow around a circular cylinder for Re num-bers ranging from 60 to 180, using the discrete-vortex method. They were ableto reproduce the Reynolds number dependence of the Strouhal number as t h a tobtained by t h e careful experiments of Williamson (1989) (see Fig. 1.9 for t h elatter experimental d a t a ) . Comparison is reproduced here in Table 5.3. Table 5.3 Strouhal numbers for Re = 60-180 computed by Stansby and Slaouti (1993) by the discrete vortex method. Experimental data from Williamson (1989). Re 60 100 140 180 Computed 0.139 0.166 0.180 0.192Strouhal number Experimental 0.135 0.164 0.180 0.191Strouhal number T h e force coefficients including the skin-friction drag and the skin-frictionlift obtained by Stansby and Slaouti (1993) are shown in Fig. 5.22 for the testedlowest and highest Re numbers, Re = 60 and 180. T h e mean drag coefficient valuesare in very good agreement with those obtained by the N . - S . solutions presentedin Fig. 5.7. Also, it may be mentioned t h a t Stansby and Slaouti m a d e a detailedcomparison between their results and the results obtained from the finite-elementand the spectral methods and found an agreement within 2-4%. Regarding the lift coefficient, no experimental d a t a are available for suchsmall Re numbers. Comparison of the results with those found from the previouslymentioned methods show, however, t h a t the agreement is within 10-12% (Stansbyand Slaouti (1993)).
    • 242 Chapter 5: Mathematical and numerical treatment of flow around a cylinder 3 Re = 6 0 2 , drag 1 skin friction drag 0 r-=3"-H, ,«=*- 20 30 40 / 50 60 70 -1 skin friction lift lift -2 W/rn a) -3 Re= 180 Figure 5.22 Force variation with time computed from the vortex method. r0 is the cylinder radius. Stansby and Slaouti (1993).5.2.3 Application areasSteady current W h e n Re > 300, the flow becomes three-dimensional (Section 1.1). Insuch situations, the implementation of the vortex method in the way as describedin the preceding paragraphs may not be entirely correct. To account for theeffects of three-dimensionality of t h e flow, t h e concept of circulation reduction hasbeen introduced in the calculations (Sarpkaya and Shoaff, 1979). Discrete vortexmodels show t h a t the concentrated vortices in the wake contain about 80% of theshed vorticity, while experiments show t h a t this figure is around 60% (Sarpkayaand Shoaff, 1979). A model of circulation reduction basically seeks to dissipatevorticity so t h a t the 20% more reduction in circulation can be realized in the
    • Discrete vortex methods 243calculations. Apparently, this concept worked well and gave good agreement withthe experiments (see also Sarpkaya, 1989). In offshore-engineering practice the Reynolds number is rather high and thesurface roughness may be rather large, therefore t h e flow is normally postcritical.Special vortex methods have been developed to handle such situations, Smithand Stansby (1989) a n d Yde og Hansen (1991). In Smith and Stansbys work,the turbulent flow is simulated in a thin boundary region around the cylinder bysuperimposing random walks on the convection of point vortices in this region.In the calculation of r a n d o m walks, t h e molecular viscosity, v, is replaced by aneffective viscosity, ve, which is equal to ve = v + UT in which wp is the turbulenceviscosity. In the model, UT is determined from the vorticity distribution throughan algebraic turbulence model. i — i — i — i — i — i — i — i — i — i — [ — i — i — i — i — i — i — i — i 0 2 4 6 x/D 8 Figure 5.23 Vorticity field obtained through the cloud-in-cell vortex method; e/D = 0.4 in which e is the gap between the pipe and the bed. Sumer et al. (1988). In the work of Yde and Hansen (1991), on the other hand, a turbulentboundary-layer model (based on Freds0es wave boundary layer model (1984),which assumes a logarithmic velocity distribution in the b o u n d a r y layer) has beenincluded. T h e key point in Yde and Hansens method is t h a t the discrete vorticesare introduced at the "centroid of the vorticity" in the boundary layer. To pinpointwhere these points lie across the boundary layer thickness, the boundary-layer
    • 244 Chapter 5: Mathematical and numerical treatment of flow around a cylindercalculation needs to be performed at each time step, to get the boundary-layerthickness. T h e model is capable of giving the Reynolds number dependence andthe roughness dependence in the transcritical flow regimes through the assumedlogarithmic velocity distribution in the b o u n d a r y layer. T h e vortex methods have been implemented quite extensively in variousareas of fluid engineering, covering from offshore to aerospace-engineering applica-tions, such as flow around multiple cylinders (Skomedal, Vada and Sortland (1989),Yde and Hansen (1991)), oscillatory flow around cylinders (see next Section), flowaround arbitrary shaped and sharp edged bodies (Scolan and Faltinsen, 1994), flowaround a pipeline over a scoured bed (Sumer, Jensen, Mao a n d Freds0e, 1988), ton a m e b u t a few. Fig. 5.23 illustrates the vorticity field around a pipeline over aplane bed (Fig. 5.23a) and a scoured bed (Fig. 5.23b) obtained by cloud-in-cellvortex method. In this latter simulation, the vortices are released steadily into theflow from t h e boundaries, namely the pipe surface and the bed. T h e strength ofthese vortices are calculated in such a way t h a t the zero normal velocity and zeroslip conditions are satisfied together on the pipe surface and also t h a t the zeronormal velocity condition is satisfied on the bed.O s c i l l a t o r y flows a n d w a v e s T h e vortex methods have been implemented widely for prediction of flowsaround cylinders subject to waves. Stansby and Dixon (1983) extended Chorins(1973) method so as to cover the case of oscillatory flows. Later, similar workswere carried out by Stansby and Smith (1989), Skomedal et al. (1989) and G r a h a mand Djahansouzi (1989). Fig. 5.12 shows a comparison between the results of Stansby and Smith(1989) and those of other methods (namely, Justensens (1991) N . - S . solution andthe results of the asymptotic theory described in Example 4.3) and the exper-iments. T h e agreement between the discrete vortex method, the N.-S. solutionand the asymptotic theory appears to be rather good. T h e vortex-method resultsagree quite well with t h e experiments except the KC range between 1 and 2.5.This may be linked to the 3D Honji vortices and transitional flow regimes (b andc in Fig. 3.15) experienced in 1 < KC < 2.5 for j3 = 1035. In the previously mentioned studies, the Reynolds number was kept rathersmall to satisfy the laminar-flow conditions. As noted in the preceding section,special vortex methods have been developed to cope with the situations where thepostcritical flow regimes prevail with the boundary layer being partially or com-pletely turbulent; Hansen, Yde and Jacobsen (1991) used t h e algorithm presentedin Yde and Hansen (1991) to investigate the flow around single and multiple cylin-ders subject to unsteady and oscillatory flows. Two, four and eight cylinders wereinvestigated with Re = 10 5 — 5 x 10 6 and ks/D = 0 — 30 x 10~ 3 and with variousangles of attack. Valuable information was obtained with regard to, among others,the influence of spacing between the cylinders on loading. Fig. 5.24 illustrates howan impulsively-started flow develops around two cylinders in t a n d e m arrangement.
    • Discrete vortex methods 245 Figure 5.24 Simulated impulsively-started flow around two cylinders in tan- dem. Hansen et al. (1991). T h e vortex methods have been used for the case of orbital flow as well,Stansby and Smith (1991) and Stansby (1993). T h e latter authors conducted thediscrete-vortex simulations for low KC numbers and low /3 numbers (see Exam-ple 4.5 for a full discussion of the forces on cylinder in orbital flows at low KCnumbers). Fig. 5.25 shows the steady streamlines, averaged over a number ofcycles for various values of ellipticity, E, and the KC number, taken from Stansby(1993). While, for zero ellipticity (i.e., the planar oscillatory flow), the streamlinesclearly illustrate the steady streaming p a t t e r n studied earlier in Section 3.6 (Fig.3.38), this p a t t e r n is disrupted with increasing E, and eventually degenerates intoa steady, recirculating streaming in the case of circular orbital motion (for E = 1).Fig. 5.26, on t h e other hand, shows the vorticity picture with the backgroundstreamlines as obtained in Stansby and Smiths study (1991). Both Stansbys andStansby and Smiths works show a substantial reduction in the inertia force, infull accord with the previously mentioned observations (Example 4.5). Stansby(1993) gives also numerically obtained drag coefficients in addition to the inertiacoefficient data.
    • 246 Chapter 5: Mathematical and numerical treatment of flow around a cylinder E=0.00 KC=0.50 E=0.25 KC=0.50 E=1.00 KC=0.50 E=0.00 KC=1.50 E=0.25 KC=1.50 E=1.00 KC=1.50 Figure 5.25 Steady streamlines for orbital flow, averaged over cycles 16-20 for E > 0 and over cycles 10-14 for E = 0. Stansby (1993).
    • Discrete vortex methods 247 . (0.125) V J-> Figure 5.26 Streamline and vorticity contours for uniform, circular, onset flow with KC = 1.5, a t various t/T, shown by t h e number in t h e cylinder. T is the wave period. T h e arrow on the streamline shows t h e incident flow direction. T h e green area shows vorticity of clockwise rotation, the red area vorticity of anticlockwise ro- tation. Stansby and Smith (1991) with permission - see Credits.
    • 248 Chapter 5: Mathematical and numerical treatment of flow around a cylinder5.3 Hydro dynamic stability approach T h e formation of vortex shedding behind a cylinder may be viewed as aninstability of the flow in the wake. T h e instability emerges because t h e presence ofthe wake behind t h e cylinder introduces two shear layers as sketched in Fig. 5.27.Shear layers are known to be unstable, a n d t h e familiar hydrodynamic stabilityanalysis can be employed to predict the frequency and the spacing of t h e vortexshedding. Such an analysis has been carried out by Triantafyllou et al. (1986 and1987) for a circular cylinder. T h e following paragraphs will summarize this work. Assuming a two-dimensional and parallel flow with the velocity componentsgiven by u = U(y) + u (5.69) v = 0 + v (5.70)and the pressure p = P + p (5.71)and writing the infinitesimal disturbances introduced in the velocity components,namely u and v, in terms of a stream function ip as •--£ (-)and furthermore neglecting the quadratic terms, the N . - S . equations and t h e con-tinuity equation (Eqs. 5.42-5.44) lead t o t h e so-called Orr-Sommerfeld equation(Schlichting, 1979, p. 460): (kU-u>)(<f>" -k2<j>)-W"<t> = 2 4 = -iu(<t>"" -2k <j>" + k 4>) (5.74)Here <j> is defined as the amplitude in the stream function of t h e disturbance flow V>(x, y , t) = ^ K « — * " > (5.75)in which k is t h e wave number, u is t h e angular frequency of t h e introduceddisturbance, and i is the imaginary unit ( = %/^T). Eq. 5.74 is the basic equation
    • Hydrodynamic stability approach 249for the stability analysis. W h e n the mean flow U(y) is specified, the solution ofthe equation (i.e., t h e eigen solutions) give u and k: UJ = ujr + i u>i (5.76) K — /Cj- ~~ Z Kj (5.77)If Ui is positive, it will represent the growth rate of the introduced disturbancein time (cf. Eq. 5.75), otherwise it will represent the decay rate. Likewise, fc;expresses the growth rate in space of the disturbance when it is negative and thedecay rate otherwise. Triantafyllou, Triantafyllou and Chryssostomidis (1986, 1987) consideredthe inviscid version of t h e Orr-Sommerfeld equation, known as the Rayleigh equa-tion: {kU-ui) {<t>" -k2<j>)-kU"<j> = 0 (5.78)with the velocity profile U{y) given by U(y) (5.79) Uo 1 - A + A tanh «[(£)in which Uo is the mean flow velocity as y —» oo, and A, a and b are curve-fittingparameters determined from the actual, measured mean velocity profiles (see Fig.5.27 for definition sketch). Shear layer U„ Shear layer Figure 5.27 Velocity profile considered in the hydrodynamic stability ana- lysis.
    • 250 Chapter 5: Mathematical and numerical treatment of flow around a cylinder T h e so-called parallel flow assumption has been m a d e in Triantafyllou etal.s study. Namely, the mean flow is assumed to vary gradually with the distancex, so t h a t locally the instability properties of t h e wake can be adequately repre-sented by the instability properties of a parallel flow (namely, a constant velocityprofile extending over an infinite x distance) having the same mean velocity profileas the local wake section considered. Hence, whether the flow is unstable has beendetermined as function of the distance x. Triantafyllou et al. did the calculations for three families of U(y) profiles.The first two, one for Re = 30 and the other for Re = 56, were taken fromKovasznays (1949) measurements. T h e third one, taken from Cantwell (1976),corresponded to a turbulent wake with a Re number equal to 140.000 ("pseudo-laminar" flow calculations). Although Triantafyllou et al. considered the inviscidOrr-Sommerfeld equation, it is clear t h a t the Reynolds number dependence isintrinsic in the analysis through t h e considered velocity distributions. Re 5 10 - 10 3 i Stable 10 - 2 Unstable v 10 - o 6 o o o o 10 Stable •j i i i 11 n i -i 0.1 10 x/D Figure 5.28 Stability diagram for flow past a cylinder by Triantafyllou et al. (1987). Triangles: Absolute instability. Circles: Convective instability. Fig. 5.28 displays the results of Triantafyllou et al.s analysis. In the figure,the "unstable" region is the region of absolute instability while the "stable" region
    • Hydrodynamic stability approach 251is that where there is only convective instability (i.e., a wave t h a t grows as ittravels; when the disturbance is convected away, however, the oscillations willeventually die out). Fig. 5.28 shows the following. 1) T h e flow is unstable (i.e.,vortex shedding occurs) if Re > 40. This is because when t h e Reynolds numberbecomes so large (larger t h a n about 40), the dissipative (or damping) action ofviscosity then becomes relatively weak. This leads to the change in the mode offlow in the form of vortex shedding. Regarding the critical value of Re, namelyRe = 40, this value is in good agreement with experiments (see Section 1.1).2) Furthermore, it is seen t h a t the streamwise extent of t h e region of instabilitydecreases with increasing Re. Triantafyllou et al. related this to t h e so-calledformation region, which determines the frequency of vortex formation. Apparently,the results regarding the size of the region of instability are consistent with thecorresponding dimensions reported for the formation region (Triantafyllou et al., 1986 and 1987). At the x-sections where there is instability, the corresponding values of urand kr would give the frequency and the spacing of the vortex shedding, respec-tively:T h e results obtained by Triantafyllou et al. (1987) regarding the above quantitiesare summarized in Table 5.4. As seen, the Strouhal frequencies obtained by meansof the stability analysis agree remarkably well with the experimental d a t a givenin Fig. 1.9. Table 5.4 Frequency and spacing of vortex shedding obtained through the stability analysis of Triantafyllou et al. (1987). A Re x/D UrD/Uo krD *< = € D 56 2.0 0.83 1.1 0.13 5.7 3.5 0.83 1.45 0.13 4.3 5.0 0.83 1.2 0.13 5.2 8.0 0.83 1.05 0.13 6.0 20.0 0.83 0.90 0.13 7.0 1.4 x 10 5 1.0 1.3 2.2 0.21 2.9 2.0 1.3 1.9 0.21 3.3
    • 252 Chapter 5: Mathematical and numerical treatment of flow around a cylinder It may be noted t h a t Triantafyllou et al. (1987) developed a model of thewake, based on t h e results of their instability analysis, which is able to obtaingood estimates of the steady and unsteady forces on t h e cylinder. Finally, it may be mentioned t h a t a similar analysis, b u t only for a laminarwake and with a different velocity profile expression, was undertaken by Nakaya(1976) with some limited results, indicating t h a t the wake flow may become un-stable for Re number above a value of about 40-50. t = 0.00 / U t = 0.25/U t = 0.30X/U t = 0.35^/U t = 0 . 4 0 A./U 0 0.5 1.0 1.5 2.0 ^ """ Figure 5.29 Instability of shear layer. Rosenhead (1931). In t h e context of hydrodynamic stability, it would be interesting to recallsome of t h e previously mentioned information given in Section 5.1 in relation tothe direct solution of N . - S . equations. T h e knowledge on hydrodynamic stabilityregarding the flow around a cylinder may be obtained directly from the solutionof N . - S . equations. In fact, Fromm and Harlows (1963) calculations did indicatethat, for Re < 40, the flow around a rectangular cylinder remained stable (i.e.,no shedding developed) after the introduction of a small perturbation in the formof an artificial increase in the value of t h e vorticity just in front of the cylinder.For Re > 40, however, their calculations showed t h a t the flow became unstable tosuch small perturbations; they reported t h a t within a fairly short time after theintroduction of the perturbation, the shedding process began to occur. Apparently,to achieve the flow instability, introduction of small artificial perturbation in one
    • Hydrodynamic stability approach £53form or another is a common practice used in t h e numerical solution of t h e N . - S .equations, unless t h e round-off errors in t h e calculations excite t h e vortex sheddingprocess (Table 5.1). tu. u a U 0.000; 0.0035 0.184; 0.0258 0.384; 0.0968 K /J % /J S 0.584 ; 0.2370 0.784; 0.3503 1.184; 0.5470 Figure 5.30 Vortex street formation with h/a = 0.281, A = —0.0250a, 7 = (tanh7rft/a), n = 2 1 , and A t = 0.004a/l7. Abernathy and Kronauer (1962). U: the mean horizontal velocity of translation of the vortex system.
    • 254 Chapter 5: Mathematical and numerical treatment of flow around a cylinderI n s t a b i l i t y o f t w o parallel c o n c e n t r a t e d s h e a r layers Another approach to study the instability of the wake flow is to assumethat the wake flow may be simulated by two parallel shear layers, where the shearis concentrated into a single step in flow velocity (rather t h a n the more smoothdistribution as given by Eq. 5.79). Regarding the instability of shear layers in general, the work in this areadates back as early as 1879; earlier studies of Rayleigh (1879) showed t h a t parallelshear flows are unstable. Rosenhead (1931) studied the instability of a shearlayer with an infinitesimal thickness using the vortex method. Rosenheads studyshowed t h a t 1) t h e shear layer is unstable t o small disturbances, 2) t h e initiallysinusoidal disturbance grows asymmetric, and 3) the vorticity in t h e shear layereventually concentrates in vortices (Fig. 5.29). T h e frequency associated with theaforementioned shear-layer instability could not be predicted through t h e methodof Rosenhead since the effect of diffusion was not taken into consideration; thisfrequency is known to depend on the m o m e n t u m thickness of the shear layer (Hoand Huerre, 1984). T h e m e t h o d of Rosenhead (1931) was later adopted by A b e r n a t h y and Kro-nauer (1962) to study the instability of two parallel shear layers, simulating thewake.flow behind a bluff body. This study was successfull in demonstrating thatthe vorticity in the shear layers concentrates into vortices and further that the vor-tices are eventually arranged in a staggered configuration, reminiscent of K a r m a nstreet (Fig. 5.30). Similar to Rosenheads study, the frequency or the spacingassociated with the instability could not be obtained by t h e applied method. A b e r n a t h y and Kronauer studied in detail the instability of the two shearlayers for various values of t h e parameter h/a in which a is t h e wave length of theinitial disturbance and h the distance between the shear layers. They found t h a tthe p a t t e r n of vortex street formation did not change with h/a. They observed,however, t h a t h/a = 0.28 is the smallest shear-layer spacing for which only twoclouds form per wave length. This value coincides with the value obtained byK a r m a n (1911 and 1912) as the stability condition for two infinite rows of pointvortices in a staggered configuration where h is the spacing of the two arrays ofvortices and a the distance between the vortices on t h e same array.Example 5.3: Karmans stability analysisSingle infinite row: For reasons of simplicity, first consider an infinite row of vortices located atthe points 0, ± a , ± 2 a , ...., each with strength K (Fig. 5.31). T h e complex potential of 2n + 1 vortices nearest the origin (including theone at t h e origin) is
    • Hydrodynamic stability approach 255 y,: -6 e e e- -2a -a 0 2a Figure 5.31 A single row of vortices. wn = ire In 2 + iK,n(z — a) + ... + ireln(.z — na) + inn{z + a) + ... + ireln(2 + na) (5.81)in which, for example, the term ireln(z — a) represents the contribution to wnof the vortex located at z = a + «0 = a (Milne-Thomson, 1962, Section 13.71).Combining the terms ""»<--5H-i& a a 2 + l reln -a (2V)...(nV) (5.82)and omitting the second term (because it will not contribute to the velocity, sinceit is constant): wn = IK, in < — 1 (5.83) 22a2From the identity(Abramowitz and Stegun, 1965, Formula 4.3.89), the complex potential in Eq.5.83, when n —> oo, will be w = i/clnl sin (5.85) •(¥) T h e complex velocity at the vortex z = 0 induced by t h e remaining vorticesof the infinite row is
    • 256 Chapter 5: Mathematical and numerical treatment of flow around a cylinder / dw dz / z=o -£{iKHH^-))-iKlnz}z=0 . /W •KZ 1 — IK I — COt I = 0 (5.86) a a z)z=oHence, t h e vortex at z = 0 is at rest, and therefore all the vortices are at rest,meaning t h a t the row induces no velocity in itself.Two infinite rows in a staggered configuration. Karmdn vortex street In order t o consider t h e two shear layers in t h e downstream wake, we nowconsider two infinite rows of vortices in a staggered configuration at time t = 0(Fig. 5.32). T h e vortices in the rows have equal strengths, namely K, b u t oppositerotation. Also, note t h a t t h e ones in t h e upper row are at points ma + j i h (m =0, ± 1 , ± 2 , . . . ) and those in t h e lower row at the points (n + | ) a — ih (n =0, ± 1 , ± 2 , . . . . ) . B -0- h/2 —-Q h/2 _i__Q. -e— —--e- -H- a/2 a/2 Figure 5.32 Two infinite row of vortices. The complex potential for this arrangement of vortices at time t = 0 istherefore w = in In + z( — « ) l n -(i(-i+T)) (5.87)
    • Hydrodynamic stability approach 257in which the first term is the contribution of the upper row, while the second termis that of the lower row (see Eq. 5.85). The velocity of the system may be calculated as follows. The velocity of thevortex at z = | a — ih (Vortex A): dw dz A. (5.88) dz •(:(-?))(on taking only the term in w associated with the upper row, as the lower rowdoes not induce any velocity in itself, as discussed in the preceding paragraphs).Hence dw .KIT —i— cot (5.89) dz -iih a 2 ~~o7)Using, tanhx = — it&n(ix) (5.90)(Abramowitz and Stegun, 1965, formula 4.5.9), Eq. 5.89 dw K7T /""« — tannl — ) (5.91) dz -iihThis indicates that the vortex moves in the x-direction with this velocity, and sodo all the vortices of the lower row, meaning that the lower row advances withvelocity KX , /whs V = —tanh — (5.92) a aJand, likewise, the upper row advances with the same velocity. The stability analysis. The procedure of Karmans stability analysis is basi-cally as follows: 1) displace the vortices slightly according to a periodic disturbanceand 2) determine whether the displacement of vortices ever grow (instability) orotherwise (stability). The governing equation used for the analysis is simply theequation of motion for any one of the vortices: dz (5.93)in which z = x — iy, the conjugate complex of z, the location of that particularvortex, and u — iv is the complex velocity induced by all the other vortices at thatpoint.
    • 258 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Now, first, move t h e vortices slightly with t h e following displacements zm = 7 cos(m</J) (5.94) : 7 cos (n+ l) (5.95)in which zm and zn are the displacements for the upper and lower vortices, re-spectively, 7 and 7 are small complex numbers, and <j> is 0 < <f> < 2n. Second, work out the velocity of, for example, the vortex at z = 0 + ih attime t = 0 (namely, Vortex B). T h e contributions t o this velocity from t h e vorticescorresponding to ± m in the upper row, will be dw d . . • iv = — — = —— { z/cln z — I am + —- + z„ dz dz ih + IK In am+ — + z_m) (- z=0+^ + z0 + • zo — Zm, — ma za — z~m + ma or expanding by the binomial theorem a n d retaining the first powers of 20, zm, Z-m zm + **—m 2z0 m ~ z (5.96)and those from the vortices coresponding to —n — 1 and n in the lower row dw 1 iv = — = in -TT- + dz z0 -zn-{n + l / 2 ) a + ih z0 - z i n _ x + (n + l / 2 ) a + ihor, by the binomial expansion and retaining the first powers of ZQ, z_n_x, zn Zp ~ Z i n _ ! ZQ -Zn (n + l / 2 ) o + ih]2 [(n + l / 2 ) a - ih}2 1 1 z I z (5 97) V (n + l/2)a-ih (n + l / 2 ) a + ih From Eqs. 5.95 and 5.96, and using Eqs. 5.93 and 5.94, t h e total velocity of thevortex is found as
    • Hydrodynamic stability approach 259 ir-^ 2ni 7(1 — cos(mi^)) ^— a2 m2 m=l °° 2ni [7 - 7 cos((n + 1/2)0)1 (n + 1/2)2 - k: n=0 [(n + 1/2)2 + P ] «; 2fca E ^ ( n +1/2) + F n=o 2 (5 98) vhich k =- (5.99) a Third, apply the equation of motion (5.93) for the considered vortex (VortexB) for which dz/dt is — - v + — dt ~ dt =V + Jt (^ cos (°^)) = V + J (5-10°)and, from Eqs. 5.93, 5.98 and 5.100, one gets 2KI ^ 7(1 - cos(m<t>)) ^ 2<ci[7 - 7 cos(n + 1/2)61] [(n + 1/2)2 - k2} 2 m ° ^ 1 " h , a2[(n + l/2)2 + k2]2 n=0Using the identity £(, + l/2) 2 + fc 2= 2>nh^ ^102)(Gradshteyn and Ryzhik, 1965, formula 1.421.2), and recalling Eq. 5.92, theequation of motion (5.101) will be S = ^ ( A 7 + C7<) (5,03)in which A and C are
    • 260 Chapter 5: Mathematical and numerical treatment of flow around a cylinder •^ 1 — cost m(j>) 4-^ (n+)2—k2 x J (5104) - £ - ^ - - £ [(•.+«•+»]• cf-K^H--l.-KyW (5,05) ... [(» + !)+"] For a vortex in the lower row, replacing K with — K and interchanging 7 and7, the counterpart of Eq. 5.103 is obtained as f = ^W + CT) (5.106) The fourth step in the analysis is to solve Eqs. 5.103 and 5.106 to get 7and 7, the two unknowns of the problem. For this, differentiate Eq. 5.103 withrespect to t: cPj 2in ( ,d-y ^.dfThe conjugate of the above equation is thenand using Eqs. 5.103 and 5.106, the following differential equation is obtained for $ - £ ( A - C h =0 (5.109)A similar equation may be obtained also for 7. Now, a trial solution for 5.109 is 7 = Gexp(^A*) (5.110)which yields A2 - (A2 - C2) = 0 (5.111)The discriminant of this second degree equation is A = A(A2-C2) (5.112)if A > 0, A will be real, therefore the motion will be unstable. Now, consider the case when <j> = 7r, which gives the maximum disturbance(Eqs. 5.94 and 5.95). In this case, from Eq. 5.105, C becomes nil, therefore fromEq. 5.112 A = 4A2 (5.113)
    • Hydrodynamic stability approach 261which is always positive, meaning that the motion is always unstable, unless A = 0.The latter condition, from Eq. 5.104, reads m m=l n=0 [(" + 2) + k IThe first series in the preceding equation is ^l-cos(Trm) 2 2 2 w2 w2 /f.,lrs p 32 52 - g 4 m=l(Gradshteyn and Ryzhik, 1965, formula 0.234.2), and the second series, by differ-entiation of Eq. 5.102 with respect to k, v (»+i)2-*2 «2 f5116) 2 2 2 h[{n+l) +k} 2cosh2(fc7r)and therefore Eq. 5.114 will be ,2 ^ -7 =0 (5.117) 4 2cosh2(fc7r) V ;yielding kit- = 0.8814, or h = 0.281a (5.118)As a conslusion, the motion (or the arrangement of vortices in Fig. 5.32) is alwaysunstable unless the ratio h/a has precisely this value, namely 0.281. For a more detailed discussion of this topic, reference may be made to Lamb(1945, Article 156). Lamb further shows that, for all values of <f> from 0 to 27r, thearrangement is stable for h/a = 0.281. Also, as another stability problem, Lambdiscusses the case of symmetrical double row, and shows that this arrangement isalways unstable.Instability of shear layer separating from cylinder Experiments show that an instability develops in the shear layer separatingfrom the cylinder, where the shear layer rolls into small vortices, when Re becomeshigher than about 2000 (Bloor (1964), Gerrard (1978), Wei and Smith (1986),
    • 262 Chapter 5: Mathematical and numerical treatment of flow around a cylinder Figure 5.33 Instability of shear layer separating from the cylinder, where the shear layer rolls into small vortices.Kourta, Boisson, Chassaing and Minh (1987) and Unal and Rockwell (1988)).Fig. 5.33 illustrates the small-scale vortices formed as a result of this instability. T h e instability waves corresponding to these small-scale vortices are oftencalled transition waves. T h e frequency of these waves, / j , is considerably highert h a n the frequency of vortex shedding / „ . Braza, Chassaing and Minh (1990) has studied the aforementioned insta-bility by the numerical simulation of the flow in the range Re = 2 X 10 3 — 10 4by solving the two-dimensional N . - S . equations. Although the transition mecha-nism leading to the transition-waves instability is analogous to t h a t generating theinstability of a free shear layer (Ho and Huerre, 1984), there may be an interac-tion between the transition-waves instability and the instability leading to vortexshedding. Braza et al., among other issues, examined this interaction. Fig. 5.34illustrates t h e velocity field together with the schematic representation of vorticescorresponding to the presented velocity field for Re = 3000 obtained in Brazaet al.s study. Fig. 5.35 compares the numerically obtained d a t a on the ratio offt/fv with experiments. From the figure, it is seen t h a t while ft/fv is about 5 forRe = 2 x 10 3 , it becomes about 18 when Re S 3 x 10 4 .3-D instabilitySteady current: Another instability in relation to t h e flow around cylinders is the onset ofthree-dimensionality for the Reynolds numbers larger t h a n about 200, see Section1.1. This phenomenon has been investigated numerically by Karniadakis and
    • Hydrodynamic stability approach 26S Figure 5.34 (a) Velocity field, (b) Schematic representation of main ( M ) and secondary ( 5 , T) vortices in the near wake. Re = 2000. Braza et al. (1990).Triantafyllou (1992) by direct simulation of the N . - S . equation in the range ofRe, 175 < Re < 500. Karniadakis and Triantafyllous calculations showed thatwhile, for Re = 175, the flow remained stable, the instability set in (i.e., the three-dimensionality occurred) when the Reynolds number is increased to Re = 225,being consistent with the observations. Figs. 5.36 and 5.37 show time series of t h e streamwise and spanwise com-ponents of the velocity for the previously mentioned Re numbers. T h e spanwisecomponent of the instantaneous velocity, w, may be used as a measure of thethree-dimensionality. From the time series of w presented in Figs. 5.36 and 5.37,it is seen t h a t , while a noise, initially introduced into the flow, dies out for the caseof Re = 175, it apparently grows a n d eventually settles for a constant amplitudein the case of Re = 225.
    • 264 Chapter 5: Mathematical and numerical treatment of flow around a cylinder I I 1.0 D C © * 0.5 ~ n © + I I 3.0 3.5 4.0 4.5 log10(Re) Figure 5.35 Ratio of the transition wave frequency over Strouhal frequency versus Reynolds number.®, +:Bloor (1964); •, Gerrard (1978); o, Kourta et al. ( 1 9 8 7 ) ; © , 3 , Wei and Smith (1986);ABraza et al.s (1990) direct numerical simulation. Adapted from Braza et al. (1990). Further to their direct simulation at Re = 175 and 225, Karniadakis andTriantafyllou (1992) have studied the transition to turbulence by conducting the3-D simulations also for Re numbers Re = 300, 333 and 500. Another three-dimensional stability analysis has been carried out by Noackand Eckelmann (1994). using low-dimensional Galerkin method. Their key resultsare as follows: 1) T h e flow is stable with respect to all perturbations for Re < 54.2) While the 2-D perturbations (of the vortex street) rapidly decay, 3-D perturba-tions with long spanwise wave lengths neither grow nor decay for 54 < Re < 170.3) T h e periodic solution becomes unstable at Re = 170 by a perturbation withthe spanwise wave length of 1.8 diameters, leading to a three-dimensional periodicflow.Oscillatory flows: As seen in Section 3.1, the oscillatory viscous flow becomes unstable tospanwise-periodic vortices above a critical KC number (the Honji instability).This kind of instability was investigated analytically by Hall (1984). Subsequently,Zhang and Dalton (1995) modelled the phenomenon numerically; they obtained adefinite 3-D behaviour as regards the variation of vorticity and also they obtainedthat the sectional lift coefficient has a strong spanwise variation.
    • Hydrodynamic stability approach 265 )Re= 175| a) b) I • -i- T ! r-» i | -*-i3 0.4 0.2 0 - • mini - : : -0.2 -0.4 -U y y y y i y y y w y i ". . . i . . . i . . 0 20 40 60 80 100 0 20 40 60 80 100 120 tU0/r0 Figure 5.36 Time history of the velocity components at x/D = 1; y/D = 0.075; z = 0 and /? = 2.0. r 0 is the cylinder radius, (a) Streamwise and (b) spanwise components. Karniadakis and Tri- antafyllou (1992). Re = 225^ a) 0.6 0.4 0.01 :| iilflilUllill IP 1 , 1 , 1 1 , , ! ... 0.2 u w 0 0 u0 U„ ; -0.2 * ^ ^ ^ -0.4 -0.6 ~ . • • • • • - • < • • • • • • • • 1 • • • • 1 • • • • I - -0.01 11 lllliil lh 0 100 200 300 400 0 100 200 300 400 tU0/r0 Figure 5.37 Time history of the velocity components at x/D = 1; y/D = 0.075; z = 0; and y8 = 2.0. r 0 is the cylinder radius, (a) Streamwise and (b) spanwise components. Karniadakis and Tri- antafyllou (1992).
    • 266 Chapter 5: Mathematical and numerical treatment of flow around a cylinderREFERENCESAbernathy, F.H. and Kronauer, R.E. (1962): T h e formation of vortex street. J. Fluid Mech., 13:1-20.Abramowitz, M. and Stegun, LA. (eds.) (1965): Handbook of Mathematical Func- tions. Dover Publications, Inc., New York.Apelt, C.J. (1961): T h e steady flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 44. R. & M. No. 3175, A.R.C. Tech. Rep., Ministry of Aviation Aero. Res. Council Rep. & Memo., 1961, 28 p.Baba, N. and Miyata, H. (1987): Higher-order accurate difference solutions of vortex generation from a circular cylinder in an oscillatory flow. J. Compu- tational Physics, 69:362-396.Badr, H.M., Dennis, S.C.R., Kocabiyik, S. and Nguyen, P. (1995): Viscous oscil- latory flow about a circular cylinder at small to moderate Strouhal number. J. Fluid Mech., 303:215-232.Baker, G.R. (1979): T h e "cloud in cell" technique applied to the roll up of vortex sheets. J. Computational Physics, 31:76-95.Batchelor, G. K. (1967): An Introduction to Fluid Dynamics. Cambridge U. Press.Bloor, M.S. (1964): T h e transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 19:290-304.Borthwick, A.G.L. (1986): Comparison between two finite-difference schemes for computing the flow around a cylinder. Int. J. for Num. Meth. in Fluids, 6:275-290.Braza, M., Chassaing, P. and Minh, H.H. (1986): Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech., 165:79-130.Braza, M., Chassaing, P. and Minh, H.H. (1990): Prediction of large-scale transi- tion features in the wake of a circular cylinder. Phys. Fluids, A2(8):1461- 1471.Braza, M., Nogues, P. and Persillon, H. (1992): Prediction of self-induced vibra- tions in incompressible turbulent flows around cylinders. Proc. 2nd I S O P E Conf., San Francisco, USA, J u n e 14-19, 1992, 3:284-292.
    • References 267Bugliarello, G. (1971): Some examples of stochastic modelling for mass and mo- mentum transfer. In: Stochastic Hydraulics (Ed. Chao-Lin Chiu), Proc. 1st Int. Symp. on Stoch. Hyd., Univ. of Pittsburgh, Perm., USA, May 31-June 2, 1971, p p . 39-55.Cantwell, B.J. (1976): A flying hot wire study of the turbulent near wake of a circular cylinder at a Reynolds number of 140.000. Ph.D.-Thesis, California Institute of Technology, Pasadena, CA.Chorin, A.J. (1973): Numerical study of slightly viscous flow. J. Fluid Mech., Vol. 57, part 4, p p . 785-796.Chorin, A.J. (1978): Vortex sheet approximation of boundary layers. J. Compu- tational Physics, 27:428-442.Chorin, A.J., Hughes, T.J.R., McCracken, M.F. and MarsdenTJ.E. (1978): Prod- uct formulas and numerical algorithms. Communications on P u r e and Ap- plied Mathematics, 31:205-256.Christiansen, J.P. (1973): Numerical simulation of hydrodynamics by the method of point vortices. J. Computational Physics, 13:363-379.Crank, J. (1975): T h e mathematics of diffusion. Clarendon Press, Oxford, U.K.Davis, R.W. and Moore, E . F . (1982): A numerical study of vortex shedding from rectangles. J. Fluid Mech., 116:475-506.Dennis, S.C.R. and Chang, G.-Z. (1970): Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech., Vol. 42, part 3, pp. 471-489.Franke, R., Rodi, W. and Schonung, B. (1990): Numerical calculation of laminar vortex-shedding flow past cylinders. J. Wind Engineering and Industrial Aerodynamics, 35:237-257.Freds0e, J. (1984): Turbulent boundary layer in wave-current motion. J. Hydraulic Engineering, ASCE, 110(8):1103-1120.Fromm, J . E . and Harlow, F.H. (1963): Numerical solution of the problem of vortex street development. Phys. of Fluids, July 1963, 6(7):975-982.Gerrard, J.H. (1978): T h e wakes of cylindrical bluff bodies at low Reynolds number. Phil. Transactions of the Royal Soc. London, Series A, 288(A1354):351-382.
    • 268 Chapter 5: Mathematical and numerical treatment of flow around a cylinderGradshteyn, I.S. and Ryzhik, I.M. (1965): Table of integrals, series and products. Academic Press, N.Y. and London.G r a h a m , J.M.R. and Djahansouzi, B. (1989): Hydrodynamic damping of struc- tural elements. Proc. 8th Int. Conf. O M A E . T h e Hague, T h e Netherlands, 2:289-293.Hall, P. (1984): On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech., 146:347-367.Hansen, E.A., Yde, L. and Jacobsen, V. (1991): Simulated turbulent flow and forces around groups of cylinders. Proc. 23rd Annual O T C , Houston, TX, May 6-9, 1991, Paper No. 6577, pp. 143-153.Ho, C.-H. and Huerre, P. (1984): Perturbed free shear layers. Ann. Rev. Fluid Mech., 16:365-424.Jordan, S.K. and Fromm, J.E. (1972): Oscillatory drag, lift and torque on a circular cylinder in a uniform flow. Phys. of Fluids, 15(3):371-376.Justesen, P. (1990): Numerical modelling of oscillatory flow around a circular cylinder. 4th Int. Symp. on Refined Flow Modelling and Turbulence Mea- surements, W u h a n , China, Sept. 1990, p p . 6-13.Justesen, P. (1991): A numerical study of oscillating flow around a circular cylin- der. J. Fluid Mech., 222:157-196.Karman, T h . von (1911): Uber den Mechanismus des Widerstandes, den ein bewegter Korper in einer Fliissigkeit erfahrt. Nachrichten, Gesellschaft der Wissenschaften, Gottingen, Math.-Phys. Klasse, p p . 509-517.K a r m a n , T h . von (1912): Uber den Mechanismus des Widerstandes, den ein bewegter Korper in einer Fliissigkeit erfahrt. Nachrichten, Gesellschaft der Wissenschaften, Gottingen, Math.-Phys. Klasse, p p . 547-556.Karniadakis, G.E. and Triantafyllou, G.S. (1992): Three-dimensional dynamics and transition to turbulence in t h e wake of bluff objects. J. Fluid Mech., 238:1-30.Kawaguti, M. (1953): Numerical solution of the Navier-Stokes equations for the flow around a circular cylinder at Reynolds number 40. Jour. Phys. Soc. of J a p a n , 8(6):747-757.
    • References 269Keller, H.B. and Takami, H. (1966): Numerical studies of steady viscous flow about cylinders. In: Numerical Solutions of Nonlinear Differential Equations. (Ed. D. Greenspan), Proc. of Adv. Symp. M a t h . Res. Center, U.S. Army at Univ. of Wisconsin, Madison, May 9-11, 1966, J o h n Wiley & Sons, Inc.Kourta, A., Boisson, H.C., Chassaing, P. and Minh, H.H. (1987): Nonlinear inter- action and t h e transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 181:141-161.Kovasznay, L.S.G. (1949): Hot-wire investigation of the wake behind cylinders at low Reynolds numbers. Proc. Royal S o c , A, London, 198:174-190.Lamb, H. (1911): On the uniform motion of a sphere through a viscous fluid. Philosophical Magazine, Vol. 2 1 , 6th Series, p p . 112-121.Lamb, H. (1945): Hydrodynamics. Dover Publications, New York.Lecointe, Y. and Piquet, J. (1989): Flow structure in the wake of an oscillating cylinder. Trans, of ASME, J. of Fluids Engineering, 111:139-148.Leonard, A. (1980): Review: Vortex methods for flow simulation. J. Computa- tional Physics, 37:289-335.Maull, D.J. and Milliner, M.C. (1978): Sinusoidal flow past a circular cylinder. Coastal Engineering, 2:149-168.Milne-Thomson, L.M. (1962): Theoretical Hydrodynamics. 4. ed., Macmillan.Murashige, S., Hinatsu, M. and Kinoshita, T. (1989): Direct calculations of the Navier-Stokes equations for forces acting on a cylinder in oscillatory flow. Proc. 8th Int. Conf. O M A E , T h e Hague, T h e Netherlands, 2:411-418.Nakaya, C. (1976): Instability of the near wake behind a circular cylinder. J. Phys. Soc. of J a p a n , Letters, 41(3):1087-1088.Noack, B.R. and Eckelmann, H. (1994): A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech., 270:297-330.Obasaju, E.D., Bearman, P.W. and G r a h a m , J.M.R. (1988): A study of forces, circulation and vortex p a t t e r n s around a circular cylinder in oscillating flow. J. Fluid Mech., 196:467-494.Oseen, C.W. (1910): Uber die Stokessche Formel u n d iiber eine verwandte Auf- gabe in der Hydrodynamik. Arkiv for Mat., Astron. och Fys., 6(29):1910.
    • 270 Chapter 5: Mathematical and numerical treatment of flow around a cylinderQuartapelle, L. and Napolitano, M. (1983): Force and moment in incompressible flows. AIAA Journal, 21(6):911-913.Rayleigh (Lord Rayleigh) (1879): On the instability of jets. Proc. London Math- ematical S o c , X:4-13.Rodi, W. (1992): On the simulation of turbulent flow past bluff bodies. J. of Wind Engineering, No. 52, August, pp. 1-16.Rosenhead, L. (1931): T h e formation of vortices from a surface of discontinuity. Proc. Roy. Soc. of London, Series A, 134:170-192.Roshko, A. (1961): Experiments on the flow past a circular cylinder at very high Reynolds number. J. Fluid Mech., 10:345-356.Sarpkaya, T. (1986): Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 165:61-71.Sarpkaya, T. (1989): Computational methods with vortices - - T h e Freeman Scholar Lecture. J. Fluids Engineering, Trans. ASME, 111:5-52.Sarpkaya, T. and Shoaff, R.L. (1979): A discrete-vortex analysis of flow about sta- tionary and transversely oscillating circular cylinders. Naval Postgraduate School Tech. Report No: NPS-69SL79011, Monterey, CA.Sarpkaya, T., Putzig, C , Gordon, D., Wang, X. and Dalton, C. (1992): Vortex trajectories around a circular cylinder in oscillatory plus mean flow., J. Offshore Mech. and Arctic Engineering, Trans. ASME, 114:291-298.Schewe, G. (1983): On the force fluctuations acting on a circular cylinder in cross- flow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech., 133:265-285.Schlichting, H. (1979): Boundary-Layer Theory. 7. ed., McGraw-Hill Book Com- pany.Scolan, Y.-M. and Faltinsen, O.M. (1994): Numerical studies of separated flow from bodies with sharp corners by the vortex in cell method. J. Fluids and Structures, 8:201-230.Skomedal, N.G., Vada, T. and Sortland, B. (1989): Viscous forces on one and two circular cylinders in planar oscillatory flow. Appl. Ocean Res., 11(3):114- 134.
    • References 271Smith, P.A. a n d Stansby, P.K. (1988): Impulsively started flow around a circular cylinder by the vortex method. J. Fluid Mech., 194:45-77.Smith, P.A. a n d Stansby, P.K. (1989): Postcritical flow around a circular cylinder by the vortex method. J. Fluids and Structures, 3:275-291.Son, J.S. and Hanratty, T.J. (1969): Numerical solution for the flow around a cylinder at Reynolds numbers of 40, 200 and 500. J. Fluid Mech., Vol. 35, part 2, p p . 369-386.Spalart, P.R. and Baldwin, B.S. (1987): Direct simulation of a turbulent oscillating boundary layer. NASA Tech. Memo. 89460, Ames Res. Center, Moffett Field, CA.Stansby, P.K. (1993): Forces on a circular cylinder in elliptical orbital flows at low Keulegan-Carpenter numbers. Appl. Ocean Res., 15:281-292.Stansby, P.K. and Dixon, A.G. (1983): Simulation of flows around cylinders by a Lagrangian vortex scheme. Appl. Ocean Res., 5(3):167-178.Stansby, P.K. and Isaacson, M. (1987): Recent developments in offshore hydrody- namics: workshop report. Appl. Ocean Res., 9(3):118-127.Stansby, P.K. and Smith, P.A. (1989): Flow around a cylinder by the r a n d o m vor- tex method. In Proc. 8th Int. Conf. O M A E . T h e Hague, T h e Netherlands, 2:419-426.Stansby, P.K. and Smith, P.A. (1991): Viscous forces on a circular cylinder in orbital flow at low Keulegan-Carpenter numbers. J. Fluid Mech., 229:159- 171.Stansby, P.K. and Slaouti, A. (1993): Simulation of vortex shedding including blockage by t h e random-vortex and other methods. Int. Journal for Nu- merical Methods in Fluids, 17:1003-1013.Stokes, G.G. (1851): On the effect of the internal friction of fluids on the motion of pendulums. Trans. Cambridge Phil. S o c , Vol. 9, P a r t II, p p . 8-106.Sullivan, P.J. (1971): Longitudinal dispersion within a two-dimensional shear flow. J. Fluid Mech., Vol. 49:551-576.Sumer, B.M., Jensen, H.R., Mao, Y. and Freds0e, J. (1988): Effect of lee-wake on scour below pipelines in current. J. Waterway, Port, Coastal and Ocean Engineering, ASCE, 114(5):599-614.
    • 272 Chapter 5: Mathematical and numerical treatment of flow around a cylinderThorn, A. (1933): T h e flow past circular cylinders at low speeds. Proc. Roy. S o c , A, 141:651-669.Triantafyllou, G.S., Triantafyllou, M.S. and Chryssostomidis, C. (1986): On the formation of vortex streets behind stationary cylinders. J. Fluid Mech., 170:461-477.Triantafyllou, G.S., Triantafyllou, M.S. and Chryssostomidis, C. (1987): Stabil- ity analysis to predict vortex street characteristics and forces on circular cylinders., J. O M A E , Trans. ASME, 109:148-154.Tritton, D.J. (1959): Experiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech., 6:547-567.Unal, M.F. and Rockwell, D. (1988): On vortex formation from a cylinder. Part 1. T h e initial instability. J. Fluid Mech., 190:491-512.Wang, C.Y. (1968): On high-frequency oscillatory viscous flows. J. Fluid Mech., 32:55-68.Wang, X. and Dalton, C. (1991a): Numerical solutions for impulsively started and decelerated viscous flow past a circular cylinder. Int. Journal for Numerical Methods in Fluids, 12:383-400.Wang, X. and Dalton, C. (1991b): Oscillating flow past a rigid circular cylinder: A finite-difference calculation. J. of Fluids Engineering, 113:377-383.Wei, T. and Smith, C.R. (1986): Secondary vortices in t h e wake of circular cylin- ders. J. Fluid Mech., 169:513-533.Williamson, C.H.K. (1989): Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds number. J. Fluid Mech., 206:579-627.Yde, L. a n d Hansen, E.A. (1991): Simulated high Reynolds number flow a n d forces on cylinder groups. Proc. 10th Int. Conf. O M A E , Stavanger, Norway, J u n e 1991, l-A:71-80.Zhang, J., Dalton, C. and Wang, X. (1991): A numerical comparison of Morison equation coefficients for oscillatory flows: sinusoidal and non-sinusoidal. Proc. 10th Int. Conf. O M A E , Stavanger, Norway, J u n e 1991, l-A:29-37.Zhang, J. and Dalton, C. (1995): T h e onset of a three-dimensional wake in two- dimensional oscillatory flow past a circular cylinder. Presented a t the 6th Asian Conf. on Fluid Mech., Singapore, 1995.
    • Chapter 6. Diffraction effect. Forces on large bodies In the previous chapters, attention has been concentrated on forces on smallcylinders where the cylinder diameter, D, is assumed to be much smaller t h a n thewave length L. In this case, the presence of the cylinder does not influence thewave. In t h e case when D becomes relatively large, however, the body will disturbt h e incident waves. Consider, for example, a large vertical, circular cylinder placedon the b o t t o m (Fig. 6.1). As the incident wave impinges on the cylinder, areflected wave moves outward. On the sheltered side of t h e cylinder there will bea "shadow" zone where t h e wave fronts are bent around the cylinder, t h e so-calleddiffracted waves (Fig. 6.1). As seen, the cylinder disturbs t h e incident waves bythe generation of the reflected and the diffracted waves. This process is generallytermed diffraction. T h e reflected and diffracted waves, combined, are usuallycalled the s c a t t e r e d waves. By the process of diffraction t h e pressure around t h e body will change andtherefore the forces on the body will be influenced. It is generally accepted t h a t the diffraction effect becomes important whenthe ratio D/L becomes larger t h a n 0.2 (Isaacson, 1979). Normally, in the diffraction flow regime, the flow around a circular cylindri-cal body is unseparated. This can be shown easily by the following approximateanalysis. Consider the sinusoidal wave theory. T h e amplitude of t h e horizontalcomponent of water-particle motion at the sea surface, according to the sinusoidalwave theory, is (Eq. III. 14, Appendix III):
    • 274 Chapter 6: Diffraction effect. Forces on large bodies -*• Diffracted w a v e front s i -t i / i / 1 r 1 -• ncident i i wave / V- Ref lectf :d wave Figure 6.1 Sketch of the incident, diffracted and reflected wave fronts for a vertically placed cylinder. - H 1 a (6 1} ~ 2 tanh(fcA) in which H is the wave height, h is the water depth and k is the wave number, i.e. k (6.2) ~ L(Fig. 6.2). T h e Keulegan-Carpenter number for a vertical circular cylinder willthen be 2na KC = ~D~ <H/L) (6.3) (D/L)t&nh{kh) Obviously the largest KC number is obtained when the m a x i m u m wavesteepness is reached, namely when H/L = (H/L)m!iX. T h e latter m a y be givenapproximately as (Isaacson, 1979) : 0.14tanh(fc/i) (6.4) ^ L / maxTherefore, the largest KC number t h a t the body would experience may, from Eqs.6.3 and 6.4, be written as
    • 275 "•&M Figure 6.2 Definition sketch for a vertical circular cylinder. L KC . ^ D 6 x Waves 4- % break H / L = (H/L) m a x 2 Diffraction n 1 0 0.1 0.2 0..13 . 0 .-*~ 4 D/L Figure 6.3 Different flow regimes in the (KC, D/L) plane. Adapted from Isaacson (1979).For the KC numbers larger t h a n this limiting value, the waves will break. Eq. 6.5is plotted as a dashed line in Fig. 6.3. T h e vertical line D/L = 0.2 in t h e figure,on the other h a n d , represents the boundary beyond which the diffraction effectbecomes significant. Now, Fig. 6.3 indicates t h a t the KC numbers experienced inthe diffraction flow regime are extremely small, namely KC < 2. T h e Reynolds
    • 276 Chapter 6: Diffraction effect. Forces on large bodiesnumber, on the other hand, must be expected to be extremely large (large com-pared with O(10 3 ) in any event). From Fig. 3.15, it is seen t h a t , for KC < 2and Re > O(10 3 ), the flow will be unseparated in most of the cases. W h e n KCnumber approaches to 2, however, there will be a separation. Yet, t h e separationunder these conditions (small KC numbers) will not be very extensive (Fig. 3.15). T h e preceding analysis suggests t h a t t h e problem regarding the flow aroundand forces on a large body in the diffraction regime may be analyzed by potentialtheory in most of the situations, since the flow is unseparated. However, in somecases such as in the calculation of damping forces for resonant vibrations of struc-tures, t h e viscous effects must be taken into consideration. Obviously, u n d e r suchconditions, potential-flow theory is no longer applicable. T h e discussion given in the preceding paragraphs refers to only circularcylinders. W h e n the body has sharp corners t h e separation will be inevitable. Inthis case the viscous effects may not be negligible.6.1 Vertical circular cylinder This section will describe t h e diffraction effect, applying potential theorydeveloped by MacCamy and Fuchs (1954). T h e problem of diffraction of planewaves from a circular cylinder of infinite length has been solved analytically forsound waves (see Morse, 1986, p. 346). MacCamy and Fuchs (1954) applied theknown theory with some modifications for water waves incident on a circular pilein the case of finite water depth. T h e theory is a linear theory and the results areexact to the first order. T h e theory was initially developed by Havelock (1940) forthe special case of infinite water depth. T h e analysis given in the following paragraphs is based on the work ofMacCamy and Fuchs (1954).6.1.1 A n a l y t i c a l s o l u t i o n for p o t e n t i a l flow a r o u n d a vertical cir- cular c y l i n d e r Fig. 6.2 shows the definition sketch. T h e incident wave is coming in fromleft to right. As it impinges on the cylinder, a reflected wave moves outward fromthe cylinder, and a diffracted wave forms on the sheltered area (Fig. 6.1). Let <j>be the total potential function, defined by u,- = d<j>/dxi. T h e function <j> can befound from the following equations: T h e continuity equation (the Laplace equation):
    • Vertical circular cylinder #77 No vertical velocity at the bed: •4-=Q at z = -h (6.7) oz Bernoulli equation at the surface, where the pressure is constant (linearized): d2d> dd> , , - - 1J + 5 7 f=0 at 2 = 0 (6.8) at oz T h e velocity component normal to the surface of the body (the r-direction) — = 0 at t h e b o d y surface (6-9) Or From the linear feature of potential flow, the total potential function, <j>,can be written as the sum of two potential functions 4> = (j>l + 4>s (6.10)in which <j>i is t h e potential function of t h e undisturbed incident wave and <f>s ist h a t of t h e scattered (reflected plus diffracted) wavePotential function for the undisturbed incident wave, d>{ T h e potential function <j>i, is given by the linear theory: ^ c o s h ^ + Z O ) ^ , ^ r v 2u> cosh(fcfe) It can be seen easily that the real part of 4>i is the same as the potentialfunction given in Eq. III.6 in Appendix III. It is known that this solution satisfies V2c^ = 0 (6.12) dcf>i n =0 at 2 = 4 (6.13) Ozand ^ + 9 ^ = 0 at , = 0 (6.14)T h e quantity w in Eq. 6.11 is the angular frequency and related to k by thedispersion relation (Appendix III, Eq. III.8):
    • 278 Chapter 6: Diffraction effect. Forces on large bodies UJ2 = gkta,nh(kh) (6.15)i in Eq. 6.11 is the imaginary unit i — J—. Also, for later use, the expressionfor the surface elevation (Appendix III, Eq. III.5): " = -KS).=o = ! c o 8 ( w *"* x ) 9 (6 16) -and the velocity components (Appendix III, Eqs. III. 10 and III. 12) nents d<t>{ -KH cosh(fc(z + h)) U = C S{Ujt kx) (6 ^ = T sinh(fcfe) ° ~ -1?) dfa TTH smh(k(z + h)) . = "=a7 - ^ sinh(^) ***(«* ~k*) ( 6 - 18 )in which T is the wave period. Now, introducing the polar coordinates (Fig. 6.2), <j>i can be expressed as _ gH cosh(k(z + h)) .t,krcose e-.Wte.*rcos» (g19) 2LO cosi(kh)in which the last term from Abramowitz and Stegun (1965, Eqs. 9.1.44 and 9.1.45)can be written as eikrcose = cos(kr cos 6) + i sm(kr cos ff) OO = Jo(kr) + 2 Y^i-ty hP{kr) cos(2p0) { OO 2^(-l) p=0 p J 2 ? + i ( f c r ) cos [(2p + 1)0] OO = J0(kr) + Y,2iPJp(kr)cos(Pe) (6.20) p=iin which Jp(kr) is the Bessel function of the first kind, order p. T h e Bessel func-tions are given in tabulated forms in mathematical handbooks (e.g. Abramowitzand Stegun, 1965, Chapter 9) and also in various mathematical softwares as built-in functions (e.g. Mathsoft, 1993, Chapter 12). Fig. 6.4 gives three examples ofthe Bessel functions, namely J0, Jj and J 1 0 .
    • Vertical circular cylinder 279 1.0 0.8 0.8 Y, J0 0.4 0.4 0 6 / & 1 0 v / 1 2 / l * **-x 0 2 6 v/ft: > J n 1 0 14. H 8 / 22 v < »X -0.4 , Mo -0.4 -0.8 -0.8 Figure 6.4 Examples of Bessel functions. Ja{x), YQ(X), J{X), YI(X), Jw(x) and Y10(x). Inserting Eq. 6.20 in Eq. 6.19, t h e final form of t h e incident-wave potential ,gH cosh(fc(2 + h)) 4>i = —i X 2ui cosh(fc/i) oo Jo(kr) + J22i"Jp(kr) cos(p8) (6.21)Potential function for the scattered wave, <j>s It is assumed t h a t <f>s h a s a form similar t o Eq. 6.21. T h e particularcombination appropriate t o a wave symmetric with respect t o $ (i.e., (j>s(—9) =4>s{6)) is oo cosh.(k(z + h)) Y^ AP cos(p8) jp{kr) + iYp(kr)} e_iu" (6.22) cosh(kh) y=0in which Yf(kr) is t h e Bessel function of t h e second kind, order p (Abramowitzand Stegun, 1965. See also t h e examples given in Fig. 6.4). In Eq. 6.22, Ap(p =0, 1,...) are constants which are t o be determined from t h e b o u n d a r y conditions.Eq. 6.22 satisfies t h e Laplace equation 1 1 d24>, d2<j>s (6.23) dr2 +r - 86 + r2 7 ^ + ^ 2 = 0 ^ dff2 dz
    • 280 Chapter 6: Diffraction effect. Forces on large bodiesand the boundary conditions -p- = 0 at z - - h (6.24) ozand * £ + , & - . - - 0 . (6.25)Also, Eq. 6.22 has, for large values of r, the form of a periodic wave movingoutward in the r-direction with wave number k, and vanishing at r = oo. Thiscan be seen easily from the asymptotic form of the particular combination of theBessel functions in Eq. 6.22. This combination of Jp and Yp, known as the Hankelfunction of the first kind, H?kr) = Jp(kr) + iYp(kr) (6.26)has, for large values of r, the asymptotic form (Abramowitz and Stegun, 1965, Eq.9.2.3) 2 I— -n P~l ^ H^kr) ~ J-=-e V 4 (6.27) v V krwhich reveals that the potential function <f>3 vanishes at r = oo.The total potential function, <j> The total potential function <j> is, from Eqs. 6.10, 6.21 and 6.22, .gH cosh(fc(z + /J)) 2u> cosh(kh) oo J0(kr) + ^22i"Jv{kr)cos{p6) p=i cosh(fc(.z + h)) cosh(kh) oo x J^ Av cos (P9) [Mkr) + iYr(kr)j e~iut (6.28) p=0This function satisfies the Laplace equation (Eq. 6.6) and the boundary conditions,Eqs. 6.7 and 6.8. The only remaining boundary condition is the zero-normal-velocity condition at the surface of the body, namely Eq. 6.9. Applying this
    • Vertical circular cylinder 281condition, t h e values of t h e constants Ap(p = 0, 1,...) are determined. T h e finalform of the potential function is ,gH cosh(k(z + h)) 2LJ cosh(kh) J p=0 ( fcr >--JSr^w H^(kr0) cos(p6)e (6.29)in which t h e derivative terms are dJp(a) Jikro) = (6.30) da a=kroand rWt flJi).(fcro) = ^ W (6.31) a=fcroin which a is a d u m m y variable. In Eq. 6.29, ep is defined as 1 p= 0 (6.32) 2 p> 1 T h e Bernoulli equation (in linearized form) is used t o get t h e pressure: p = -p-£ (6.33)From Eqs. 6.29 a n d 6.33, t h e pressure on t h e cylinder surface is obtained as . pgH cosh(fc(z + h)) ^ e if _ p = i— .,,,, > —jrf, cos(p6)e (6.34) 7rfcr0 cosh(kh) ^ H^ikro) p=To reach this equation, t h e following identity is used (Spiegel, 1968, Formula24.135) Jp(a)Y;(a) - J » y » = — (6.35) T h e free-surface elevation -q can be calculated from g V dt J z=0and presumably t h e runup profiles around t h e cylinder can be worked out accord-ingly (see Sarpkaya a n d Isaacson (1981, p . 394) a n d Isaacson (1979)).
    • 282 Chapter 6: Diffraction effect. Forces on large bodies6.1.2 T o t a l force o n u n i t - h e i g h t o f c y l i n d e r Having obtained t h e wave a n d flow field a r o u n d a vertical cylinder, t h eresulting forces can easily be obtained. T h e in-line force acting on a unit height of the cylinder (Fig. 6.2) is •J.T -I p(r0d8) cos 6 (6.36)Inserting Eq. 6.34 into Eq. 6.36 and carrying out the integration and taking thereal p a r t only, t h e force is found as follows: 2pgH cosh(k{z + h)) (6.37)in which 6(kr0) = -Un-1 [Y^kr0)/Jl(kr0)} (6.38) -1/2 A(kr0)= [J 1 2 (fcr 0 ) + F 1 2 (fcr 0 )] (6.39)Here the derivatives J/(fcro) and Y^kro) are calculated in t h e same fashion as inEqs. 6.30 a n d 6.31. T h e functions A(kro) and S(kro) can be worked out, using the Bessel-function tables in Abramowitz and Stegun (1965). Figs. 6.5a and 6.5b give thefunctions A(kro) and S(kro). T h e function S(kro) represents t h e phase differencebetween the incident wave and the force, and it will be discussed later in thesection.Inertia coefficient T h e far-field velocity corresponding to the incident wave is given by Eq.6.17. From this equation, the m a x i m u m acceleration (the absolute value) is ob-tained as du nHuj cosi{k(z+ h)) (6.40) dtrn~ T smi(kh) Now, inserting Eq. 6.40 into Eq. 6.37, Fx may be expressed as 4A(kr0) d u I A- K (^o) (6.41) ir(kr0)2 — cos(wi - 6) Ot m
    • Vertical circular cylinder 283 Undisturbed incident wave at x=0 a) 0.5 1 D/L Figure 6.5 (a): The function A(kr0) in the force expression, (b): The phase function <5(fcr0) in the force expression.This equation has the same form as the Morison equation (Eq. 4.29) with thedrag omitted, namely Fx = pCM(Krl) u (6.42)(However, in Eq. 6.41, the force follows the incident wave crest (passing throughx = 0) with a phase delay equal to 8 (see Fig. 6.5b)). Hence the inertia coefficient in the case of diffraction flow regime can, fromEq. 6.41, be expressed as in t h e following 4A(fcr 0 ) C, (6.43) 7r(fcr0)2in which A(kr0) is given by Eq. 6.39. Therefore, the force Fx du i Fx = pCM{^rl) cos(o;t — 6) (6.44) I at" Ior alternatively, cosh(fc(z + h)) -pgHkD CM cos(a)i — 8) 1.45) coah(kh)
    • 284 Chapter 6: Diffraction effect. Forces on large bodies I 1 1 » 0 0.5 1 D/L Figure 6.6 The influence of diffraction on the inertia coefficient in the Mori- son equation.T h e inertia coefficient CM is plotted in Fig. 6.6 as function of kro. First of all, the figure indicates t h a t the diffraction solution approaches thevalue of 2, the plane potential-flow solution given in Eq. 4.18 (namely CM =Cm + 1 = 2), as kr0 -> 0. Secondly, CM begins to be influenced by the diffraction effect after D/Lreaches the value of approximately 0.2, in conformity with the previously men-tioned limiting value in t h e beginning of this chapter. Thirdly, the inertia coefficient decreases with increasing D/L ratio. T h ephysical reason behind this is t h a t the acceleration of flow is m a x